Axiom ax-13 2373

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 ax13b 2036). 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 (so they are effectively "distinct variables" even though no $d is present). Similarly, 𝑥 and 𝑧 cannot be the same object-language variable. Therefore, 𝑥 will not occur in the wff 𝑦 = 𝑧 when the first two antecedents hold, so analogous to ax-5 1914, the conclusion (𝑦 = 𝑧 → ∀𝑥𝑦 = 𝑧) follows. Note that ax-5 1914 cannot prove this because its distinct variable ($d) requirement is not satisfied directly but only indirectly (outside of Metamath) by the argument above.

The original version of this axiom was ax-c9 36911 and was replaced with this shorter ax-13 2373 in December 2015. The old axiom is proved from this one as Theorem axc9 2383.

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.

This axiom is mostly used to eliminate conditions requiring set variables be distinct (cf. ax6ev 1974 and ax6e 2384, for example) in proofs. In practice, theorems beyond elementary set theory do not really benefit from such eliminations, so direct or indirect application of this axiom is discouraged now. You need to explicitly confirm its use in case you see a sensible application in a niche.

After some assisting contributions by others over the years, it was in particular the extensive work of Gino Giotto in 2024 that helped reducing dependencies on this axiom on a large scale.

Although this version is shorter, the original version axc9 2383 may be more practical to work with because of the "distinctor" form of its antecedents. A typical application of axc9 2383 is in dvelimh 2451 which converts a distinct variable pair to the distinctor antecedent ¬ ∀𝑥𝑥 = 𝑦. In particular, it is conjectured that it is not possible to prove ax6 2385 from ax6v 1973 without this axiom.

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 ax13v 2374 and use only ax13v 2374 for further derivations. Thus, ax13v 2374 should be the only theorem referencing this axiom. Other theorems can reference either ax13v 2374 (preferred) or ax13 2376 (if the stronger form is needed).

This axiom scheme is logically redundant (see ax13w 2133) but is used as an auxiliary axiom scheme to achieve scheme completeness (i.e. so that all possible cases of bundling can be proved; see text linked at mmtheorems.html#ax6dgen 2133). It is not known whether this axiom can be derived from the others. (Contributed by NM, 21-Dec-2015.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-13	⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Detailed syntax breakdown of Axiom ax-13

Step	Hyp	Ref	Expression
1		vx	. . . 4 setvar 𝑥
2		vy	. . . 4 setvar 𝑦
3	1, 2	weq 1967	. . 3 wff 𝑥 = 𝑦
4	3	wn 3	. 2 wff ¬ 𝑥 = 𝑦
5		vz	. . . 4 setvar 𝑧
6	2, 5	weq 1967	. . 3 wff 𝑦 = 𝑧
7	6, 1	wal 1537	. . 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:  ax13v  2374