Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax-i12 GIF version

Axiom ax-i12 1331
 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 1335 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 1325 . . 3 wff z = x
43, 1wal 1266 . 2 wff z z = x
5 vy . . . . 5 set y
61, 5weq 1325 . . . 4 wff z = y
76, 1wal 1266 . . 3 wff z z = y
82, 5weq 1325 . . . . 5 wff x = y
98, 1wal 1266 . . . . 5 wff z x = y
108, 9wi 4 . . . 4 wff (x = yz x = y)
1110, 1wal 1266 . . 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  1335  ax12or  1336  dveeq2  1571  dveeq2or  1572  dvelimALT  1746  dvelimfv  1747
 Copyright terms: Public domain W3C validator