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Theorem axc16g-o 32418
Description: A generalization of axiom ax-c16 32377. Version of axc16g 1995 using ax-c11 32372. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc16g-o  |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem axc16g-o
StepHypRef Expression
1 aev-o 32415 . 2  |-  ( A. x  x  =  y  ->  A. z  z  =  x )
2 ax-c16 32377 . 2  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
3 biidd 240 . . . 4  |-  ( A. z  z  =  x  ->  ( ph  <->  ph ) )
43dral1-o 32387 . . 3  |-  ( A. z  z  =  x  ->  ( A. z ph  <->  A. x ph ) )
54biimprd 226 . 2  |-  ( A. z  z  =  x  ->  ( A. x ph  ->  A. z ph )
)
61, 2, 5sylsyld 58 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-c5 32368  ax-c4 32369  ax-c7 32370  ax-c11 32372  ax-c9 32375  ax-c16 32377
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664
This theorem is referenced by:  ax12inda2  32431
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