These are mostly needed for more advanced proofs, especially proofs of existence, with the existential quantifier. A handful of simple number facts have quantifiers, but their proofs don’t need these quantifier laws.

These laws all contain quantifiers, `forall`

, `exists`

, or `exists1`

for “there exists exactly one” (or “unique existence”). A few also
use `the1`

, which gives the unique value satisfying a condition, if
there is indeed exactly one such value.

Uses here of `p x`

within `{x. . . . }`

capture the same idea as
informal notations such as `∀ x. φ(x)`

found in some
textbooks.

Prooftoys form
Rearrangement:
~~forall {x. p x & q x} == forall p & forall q
exists {x. p x | q x} == exists p | exists q~~

One-way rearrangements:
~~forall p | forall q => forall {x. p x | q x}
exists {x. p x & q x} => exists p & exists q~~

Unused bound variable:
~~forall {x. A} == A
exists {x. A} == A
forall {x. A | q x} == (A | forall q)
forall {x. A => q x} == (A => forall q)
exists {x. A & q x} == A & exists q
forall {x. p x => A} == (exists p => A)~~

Forward reasoning:
~~forall {x. p x => q x} => (forall p => forall q)
forall {x. p x => q x} => (exists p => exists q)~~

Unique existence:
~~exists1 p == exists {x. p = {y. y = x}}
exists1 p == exists {x. forall {y. p y == y = x}}
exists1 p == exists {y. p y & forall {z. p z => z = y}}
exists1 p => (p x == x = the1 p)
p x & forall {y. p y => y = x} => exists1 p
(f x = the1(Q x) & exists1 (Q x)) => (Q x y == f x = y)~~

Traditional form
Rearrangement:
~~forall {x. p x & q x} == forall {x. p x} & forall {x. q x}
exists {x. p x | q x} == exists {x. p x} | exists {y. q y}~~

One-way rearrangements:
~~forall {x. p x} | forall {x. q x} => forall {x. p x | q x}
exists {x. p x & q x} => exists {x. p x} & exists {x. q x}~~

Unused bound variable:
~~forall {x. A} == A
exists {x. A} == A
forall {x. A | q x} == (A | forall {x. q x})
forall {x. A => q x} == (A => forall {x. q x})
exists {x. A & q x} == A & exists {x. q x}
forall {x. p x => A} == (exists {x. p x} => A)~~

Forward reasoning:
~~forall {x. p x => q x} => (forall {x. p x} => forall {x. q x})
forall {x. p x => q x} => (exists {x. p x} => exists {x. q x})~~

Unique existence:
~~exists1 {x. p x} == exists {x. p = {y. y = x}}
exists1 {x. p x} == exists {x. forall {y. p y == y = x}}
exists1 {x. p x} == exists {y. p y & forall {z. p z => z = y}}
exists1 {x. p x} => (p x == x = the1 p)
p x & forall {y. p y => y = x} => exists1 {x. p x}
(f x = the1(Q x) & exists1 (Q x)) => (Q x y == f x = y)~~

The proof builder will offer these laws when they apply to a term you select. Still, to understand how to use them takes a bit more than other parts of Prooftoys.

In many of these laws there is at least one occurrence of
a term like `p x`

, where `p`

is a free variable and `x`

is
bound somewhere in the statement.

Also, in some of the theorem statements there are occurrences of the
variable `A`

in the scope of a bound variable. In a form such as
`forall {x. A}`

, it is not possible to substitute for `A`

in a way
that puts the bound variable `x`

into that part of the result. Proper
substitution automatically renames the `x`

in `forall {x. A}`

to
something else if `x`

is free in the term that replaces `A`

, so the
substitution fails to put an occurrence of the bound variable into the
result.

On the other hand, in a form such as `forall {x. p x}`

, substitution
can turn the `p x`

into *any* term with a true/false value.

For example `p x`

can become `x + 0 = x`

, though it does not look like
it. The trick is to substitute for `p`

and then do a beta conversion
(axiom 4). Here `x + 0 = x`

is equivalent to `{z. z + 0 = z} x`

. So
substituting `{z. z + 0 = z}`

for `p`

and then using beta conversion
(also a form of substitution, see above) results in `x + 0 = x`

as
desired. In fact there is no need to change `x + 0 = x`

to ```
z + 0 =
z
```

. Even `{x. x + 0 = x}`

works fine and is slightly simpler, though
it may make the step a bit harder to read.

There are even more theorems like this, many dealing with specific issues in reasoning, but these classic laws are implemented in Prooftoys today.