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Theorem a16g 1744
Description: A generalization of axiom ax-16 1695. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
a16g  |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem a16g
StepHypRef Expression
1 aev 1693 . 2  |-  ( A. x  x  =  y  ->  A. z  z  =  x )
2 ax16 1694 . 2  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
3 biidd 161 . . . 4  |-  ( A. z  z  =  x  ->  ( ph  <->  ph ) )
43dral1 1618 . . 3  |-  ( A. z  z  =  x  ->  ( A. z ph  <->  A. x ph ) )
54biimprd 147 . 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 1241
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646
This theorem is referenced by:  a16gb  1745  a16nf  1746
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