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Axiom ax-13 1404
 Description: Axiom of Equality. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the left-hand side of the ∈ binary predicate. Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
ax-13 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Detailed syntax breakdown of Axiom ax-13
StepHypRef Expression
1 vx . . 3 setvar 𝑥
2 vy . . 3 setvar 𝑦
31, 2weq 1392 . 2 wff 𝑥 = 𝑦
4 vz . . . 4 setvar 𝑧
51, 4wel 1394 . . 3 wff 𝑥𝑧
62, 4wel 1394 . . 3 wff 𝑦𝑧
75, 6wi 4 . 2 wff (𝑥𝑧𝑦𝑧)
83, 7wi 4 1 wff (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
 Colors of variables: wff set class This axiom is referenced by:  elequ1  1600  el  3931
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