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Axiom ax-i12 1309
Description: Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever z is distinct from x and y, and x = y is true, then x = y quantified with z is also true. In other words, z is irrelevant to the truth of x = y. Axiom scheme C9' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases.

This axiom has been modified from the original ax-12 1311 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)

Assertion
Ref Expression
ax-i12 (z z = x (z z = y z(x = yz x = y)))

Detailed syntax breakdown of Axiom ax-i12
StepHypRef Expression
1 vz . . . 4 set z
2 vx . . . 4 set x
31, 2weq 1302 . . 3 wff z = x
43, 1wal 1253 . 2 wff z z = x
5 vy . . . . 5 set y
61, 5weq 1302 . . . 4 wff z = y
76, 1wal 1253 . . 3 wff z z = y
82, 5weq 1302 . . . . 5 wff x = y
98, 1wal 1253 . . . . 5 wff z x = y
108, 9wi 4 . . . 4 wff (x = yz x = y)
1110, 1wal 1253 . . 3 wff z(x = yz x = y)
127, 11wo 605 . 2 wff (z z = y z(x = yz x = y))
134, 12wo 605 1 wff (z z = x (z z = y z(x = yz x = y)))
Colors of variables: wff set class
This axiom is referenced by:  ax-12  1311  ax12or  1312
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