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Theorem ax-12 1311
Description: Rederive the original version of the axiom from ax-i12 1309. Note that we need ax-4 1310 for the derivation, but the proof of ax4 1326 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 1309 . . . 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 1310 . 2 (z(x = yz x = y) → (x = yz x = y))
53, 4syl6 28 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 1253   = wceq 1301
This theorem is referenced by:  dvelimfALT  1464  ax17eq  1524  hbsb4  1565  sbcom  1575  dvelimALT  1679  ax11eq  1786  ax11indalem  1790  a12stdy4  1797  a12lem1  1798
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 8  ax-ia1 98  ax-ia2 99  ax-ia3 100  ax-in2 527  ax-io 606  ax-i12 1309  ax-4 1310
This theorem depends on definitions:  df-bi 109
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