Axiom ax-12 1925

Description: Axiom of Quantified Equality. One of the equality and substitution axioms of predicate calculus with equality.

An equivalent way to express this axiom that may be easier to understand is (¬ x = y → (¬ x = z → (y = z → ∀xy = z))) (see ax12b 1689). Recall that in the intended interpretation, our variables are metavariables ranging over the variables of predicate calculus (the object language). In order for the first antecedent ¬ x = y to hold, x and y must have different values and thus cannot be the same object-language variable. Similarly, x and z cannot be the same object-language variable. Therefore, x will not occur in the wff y = z when the first two antecedents hold, so analogous to ax-17 1616, the conclusion (y = z → ∀xy = z) follows.

The original version of this axiom was ax-12o 2142 and was replaced with this shorter ax-12 1925 in December 2015. The old axiom is proved from this one as Theorem ax12o 1934. Conversely, this axiom is proved from ax-12o 2142 as Theorem ax12 1935.

The primary purpose of this axiom is to provide a way to introduce the quantifier ∀x on y = z even when x and y are substituted with the same variable. In this case, the first antecedent becomes ¬ x = x and the axiom still holds.

Although this version is shorter, the original version ax12o 1934 may be more practical to work with because of the "distinctor" form of its antecedents. A typical application of ax12o 1934 is in dvelimh 1964 which converts a distinct variable pair to the distinctor antecendent ¬ ∀xx = y.

This axiom can be weakened if desired by adding distinct variable restrictions on pairs x, z and y, z. To show that, we add these restrictions to Theorem ax12v 1926 and use only ax12v 1926 for further derivations. Thus, ax12v 1926 should be the only theorem referencing this axiom. Other theorems can reference either ax12v 1926 or ax12o 1934.

This axiom scheme is logically redundant (see ax12w 1724) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 21-Dec-2015.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-12	⊢ (¬ x = y → (y = z → ∀x y = z))

Detailed syntax breakdown of Axiom ax-12

Step	Hyp	Ref	Expression
1		vx	. . . 4 setvar x
2		vy	. . . 4 setvar y
3	1, 2	weq 1643	. . 3 wff x = y
4	3	wn 3	. 2 wff ¬ x = y
5		vz	. . . 4 setvar z
6	2, 5	weq 1643	. . 3 wff y = z
7	6, 1	wal 1540	. . 3 wff ∀x y = z
8	6, 7	wi 4	. 2 wff (y = z → ∀x y = z)
9	4, 8	wi 4	1 wff (¬ x = y → (y = z → ∀x y = z))

This axiom is referenced by:  ax12v  1926