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Axiom ax-10 1396
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 1604 ("o" for "old") and was replaced with this shorter ax-10 1396 in May 2008. The old axiom is proved from this one as theorem ax10o 1603. Conversely, this axiom is proved from ax-10o 1604 as theorem ax10 1605. (Contributed by NM, 5-Aug-1993.)

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

Detailed syntax breakdown of Axiom ax-10
StepHypRef Expression
1 vx . . . 4 setvar 𝑥
2 vy . . . 4 setvar 𝑦
31, 2weq 1392 . . 3 wff 𝑥 = 𝑦
43, 1wal 1241 . 2 wff 𝑥 𝑥 = 𝑦
52, 1weq 1392 . . 3 wff 𝑦 = 𝑥
65, 2wal 1241 . 2 wff 𝑦 𝑦 = 𝑥
74, 6wi 4 1 wff (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Colors of variables: wff set class
This axiom is referenced by:  alequcom  1408  ax10o  1603  naecoms  1612  oprabidlem  5497
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