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Theorem ax16g-o 2244
Description: A generalization of axiom ax-c16 2203. Version of axc16g 1878 using ax-c11 2198. (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
ax16g-o  |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem ax16g-o
StepHypRef Expression
1 aev-o 2241 . 2  |-  ( A. x  x  =  y  ->  A. z  z  =  x )
2 ax-c16 2203 . 2  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
3 biidd 237 . . . 4  |-  ( A. z  z  =  x  ->  ( ph  <->  ph ) )
43dral1-o 2213 . . 3  |-  ( A. z  z  =  x  ->  ( A. z ph  <->  A. x ph ) )
54biimprd 223 . 2  |-  ( A. z  z  =  x  ->  ( A. x ph  ->  A. z ph )
)
61, 2, 5sylsyld 56 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-c5 2194  ax-c4 2195  ax-c7 2196  ax-c11 2198  ax-c9 2201  ax-c16 2203
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591
This theorem is referenced by:  ax12inda2  2257
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