Axiom ax-c11n 37753

Description: Axiom of Quantifier Substitution. One of the equality and substitution axioms of predicate calculus with equality. Appears as Lemma L12 in [Megill] p. 445 (p. 12 of the preprint).

The original version of this axiom was ax-c11 37752 and was replaced with this shorter ax-c11n 37753 ("n" for "new") in May 2008. The old axiom is proved from this one as Theorem axc11 2429. Conversely, this axiom is proved from ax-c11 37752 as Theorem axc11nfromc11 37791.

This axiom was proved redundant in July 2015. See Theorem axc11n 2425.

This axiom is obsolete and should no longer be used. It is proved above as Theorem axc11n 2425. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-c11n	⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Detailed syntax breakdown of Axiom ax-c11n

Step	Hyp	Ref	Expression
1		vx	. . . 4 setvar 𝑥
2		vy	. . . 4 setvar 𝑦
3	1, 2	weq 1966	. . 3 wff 𝑥 = 𝑦
4	3, 1	wal 1539	. 2 wff ∀𝑥 𝑥 = 𝑦
5	2, 1	weq 1966	. . 3 wff 𝑦 = 𝑥
6	5, 2	wal 1539	. 2 wff ∀𝑦 𝑦 = 𝑥
7	4, 6	wi 4	1 wff (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

This axiom is referenced by:  axc11-o  37816