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Theorem a16g-o 2263
Description: A generalization of axiom ax-16 2221. Version of a16g 2048 using ax-10o 2216. (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 2259 . 2  |-  ( A. x  x  =  y  ->  A. z  z  =  x )
2 ax-16 2221 . 2  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
3 biidd 229 . . . 4  |-  ( A. z  z  =  x  ->  ( ph  <->  ph ) )
43dral1-o 2231 . . 3  |-  ( A. z  z  =  x  ->  ( A. z ph  <->  A. x ph ) )
54biimprd 215 . 2  |-  ( A. z  z  =  x  ->  ( A. x ph  ->  A. z ph )
)
61, 2, 5sylsyld 54 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549
This theorem is referenced by:  ax11inda2  2276
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-4 2212  ax-5o 2213  ax-6o 2214  ax-10o 2216  ax-12o 2219  ax-16 2221
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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