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Theorem ax-12 1335
 Description: Rederive the original version of the axiom from ax-i12 1331. Note that we need ax-4 1333 for the derivation, but the proof of ax4 1915 is nontheless non-circular since it does not use ax-12. (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 1331 . . . 4 (z z = x (z z = y z(x = yz x = y)))
21ori 617 . . 3 z z = x → (z z = y z(x = yz x = y)))
32ord 618 . 2 z z = x → (¬ z z = yz(x = yz x = y)))
4 ax-4 1333 . 2 (z(x = yz x = y) → (x = yz x = y))
53, 4syl6 27 1 z z = x → (¬ z z = y → (x = yz x = y)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 605  ∀wal 1266   = wceq 1324 This theorem is referenced by:  dvelimfALT  1911  ax11eq  1920  ax11indalem  1924  a12stdy4  1931  a12lem1  1932  ax17eq  1938 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 97  ax-ia2 98  ax-ia3 99  ax-in2 527  ax-io 606  ax-i12 1331  ax-4 1333 This theorem depends on definitions:  df-bi 108
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