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Axiom ax-14 1405
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 right-hand side of the binary predicate. Axiom scheme C13' 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. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
ax-14 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Detailed syntax breakdown of Axiom ax-14
StepHypRef Expression
1 vx . . 3 setvar 𝑥
2 vy . . 3 setvar 𝑦
31, 2weq 1392 . 2 wff 𝑥 = 𝑦
4 vz . . . 4 setvar 𝑧
54, 1wel 1394 . . 3 wff 𝑧𝑥
64, 2wel 1394 . . 3 wff 𝑧𝑦
75, 6wi 4 . 2 wff (𝑧𝑥𝑧𝑦)
83, 7wi 4 1 wff (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
Colors of variables: wff set class
This axiom is referenced by:  elequ2  1601  el  3931  fv3  5197
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