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Theorem a16g-o 2138
Description: A generalization of axiom ax-16 2096. Version of a16g 1898 using ax-10o 2091. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
a16g-o  |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem a16g-o
StepHypRef Expression
1 aev-o 2134 . 2  |-  ( A. x  x  =  y  ->  A. z  z  =  x )
2 ax-16 2096 . 2  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
3 biidd 228 . . . 4  |-  ( A. z  z  =  x  ->  ( ph  <->  ph ) )
43dral1-o 2106 . . 3  |-  ( A. z  z  =  x  ->  ( A. z ph  <->  A. x ph ) )
54biimprd 214 . 2  |-  ( A. z  z  =  x  ->  ( A. x ph  ->  A. z ph )
)
61, 2, 5sylsyld 52 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530
This theorem is referenced by:  ax11inda2  2151
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-4 2087  ax-5o 2088  ax-6o 2089  ax-10o 2091  ax-12o 2094  ax-16 2096
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
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