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Theorem ax16g-o 2266
Description: A generalization of axiom ax-c16 2225. Version of axc16g 1945 using ax-c11 2220. (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 2263 . 2  |-  ( A. x  x  =  y  ->  A. z  z  =  x )
2 ax-c16 2225 . 2  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
3 biidd 237 . . . 4  |-  ( A. z  z  =  x  ->  ( ph  <->  ph ) )
43dral1-o 2235 . . 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 1396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-c5 2216  ax-c4 2217  ax-c7 2218  ax-c11 2220  ax-c9 2223  ax-c16 2225
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622
This theorem is referenced by:  ax12inda2  2279
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