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Theorem ax11o 1700
 Description: Derivation of set.mm's original ax-11o 1701 from the shorter ax-11 1394 that has replaced it. An open problem is whether this theorem can be proved without relying on ax-16 1692 or ax-17 1416. Normally, ax11o 1700 should be used rather than ax-11o 1701, 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 1394 . 2 (x = z → (zφx(x = zφ)))
21ax11a2 1699 1 x x = y → (x = y → (φx(x = yφ))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1240 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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643 This theorem is referenced by:  ax11b  1704  equs5  1707
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