Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax-12 GIF version

Theorem ax-12 1399
 Description: Rederive the original version of the axiom from ax-i12 1395. (Contributed by Mario Carneiro, 3-Feb-2015.)
Assertion
Ref Expression
ax-12 z z = x → (¬ z z = y → (x = yz x = y)))

Proof of Theorem ax-12
StepHypRef Expression
1 ax-i12 1395 . . . 4 (z z = x (z z = y z(x = yz x = y)))
21ori 641 . . 3 z z = x → (z z = y z(x = yz x = y)))
32ord 642 . 2 z z = x → (¬ z z = yz(x = yz x = y)))
4 ax-4 1397 . 2 (z(x = yz x = y) → (x = yz x = y))
53, 4syl6 29 1 z z = x → (¬ z z = y → (x = yz x = y)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 628  ∀wal 1240   = wceq 1242 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 629  ax-i12 1395  ax-4 1397 This theorem depends on definitions:  df-bi 110 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator