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Axiom ax-i12 1332
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 1336 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 1326 . . 3 wff z = x
43, 1wal 1267 . 2 wff z z = x
5 vy . . . . 5 set y
61, 5weq 1326 . . . 4 wff z = y
76, 1wal 1267 . . 3 wff z z = y
82, 5weq 1326 . . . . 5 wff x = y
98, 1wal 1267 . . . . 5 wff z x = y
108, 9wi 4 . . . 4 wff (x = yz x = y)
1110, 1wal 1267 . . 3 wff z(x = yz x = y)
127, 11wo 608 . 2 wff (z z = y z(x = yz x = y))
134, 12wo 608 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  1336  ax12or  1337  dveeq2  1606  dveeq2or  1607  dvelimALT  1790  dvelimfv  1791
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