Axiom ax-10 1498

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-10o 1709 ("o" for "old") and was replaced with this shorter ax-10 1498 in May 2008. The old axiom is proved from this one as Theorem ax10o 1708. Conversely, this axiom is proved from ax-10o 1709 as Theorem ax10 1710. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
ax-10	⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Detailed syntax breakdown of Axiom ax-10

Step	Hyp	Ref	Expression
1		vx	. . . 4 setvar 𝑥
2		vy	. . . 4 setvar 𝑦
3	1, 2	weq 1496	. . 3 wff 𝑥 = 𝑦
4	3, 1	wal 1346	. 2 wff ∀𝑥 𝑥 = 𝑦
5	2, 1	weq 1496	. . 3 wff 𝑦 = 𝑥
6	5, 2	wal 1346	. 2 wff ∀𝑦 𝑦 = 𝑥
7	4, 6	wi 4	1 wff (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

This axiom is referenced by:  alequcom  1508  ax10o  1708  naecoms  1717  oprabidlem  5884