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Theorem ax11o 1681
 Description: Derivation of set.mm's original ax-11o 1682 from the shorter ax-11 1374 that has replaced it. An open problem is whether this theorem can be proved without relying on ax-16 1673 or ax-17 1396. Normally, ax11o 1681 should be used rather than ax-11o 1682, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.)
Assertion
Ref Expression
ax11o x x = y → (x = y → (φx(x = yφ))))

Proof of Theorem ax11o
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 ax-11 1374 . 2 (x = z → (zφx(x = zφ)))
21ax11a2 1680 1 x x = y → (x = y → (φx(x = yφ))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1224 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 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405 This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624 This theorem is referenced by:  ax11b  1685  equs5  1688
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