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Theorem ax11a2 1699
Description: Derive ax-11o 1701 from a hypothesis in the form of ax-11 1394. The hypothesis is even weaker than ax-11 1394, with z both distinct from x and not occurring in φ. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1700. (Contributed by NM, 2-Feb-2007.)
Hypothesis
Ref Expression
ax11a2.1 (x = z → (zφx(x = zφ)))
Assertion
Ref Expression
ax11a2 x x = y → (x = y → (φx(x = yφ))))
Distinct variable groups:   x,z   y,z   φ,z
Allowed substitution hints:   φ(x,y)

Proof of Theorem ax11a2
StepHypRef Expression
1 ax-17 1416 . . 3 (φzφ)
2 ax11a2.1 . . 3 (x = z → (zφx(x = zφ)))
31, 2syl5 28 . 2 (x = z → (φx(x = zφ)))
43ax11v2 1698 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:  ax11o  1700
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