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Axiom ax-i12 1375
Description: Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever is distinct from and , and is true, then quantified with is also true. In other words, is irrelevant to the truth of . 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 1379 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)

Assertion
Ref Expression
ax-i12

Detailed syntax breakdown of Axiom ax-i12
StepHypRef Expression
1 vz . . . 4  setvar
2 vx . . . 4  setvar
31, 2weq 1369 . . 3
43, 1wal 1224 . 2
5 vy . . . . 5  setvar
61, 5weq 1369 . . . 4
76, 1wal 1224 . . 3
82, 5weq 1369 . . . . 5
98, 1wal 1224 . . . . 5
108, 9wi 4 . . . 4
1110, 1wal 1224 . . 3
127, 11wo 616 . 2
134, 12wo 616 1
Colors of variables: wff set class
This axiom is referenced by:  ax-12  1379  ax12or  1380  dveeq2  1674  dveeq2or  1675  dvelimALT  1864  dvelimfv  1865
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