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Theorem a16g 1887
Description: A generalization of Axiom ax-16 1837. (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 1835 . 2  |-  ( A. x  x  =  y  ->  A. z  z  =  x )
2 ax16 1836 . 2  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
3 biidd 172 . . . 4  |-  ( A. z  z  =  x  ->  ( ph  <->  ph ) )
43dral1 1753 . . 3  |-  ( A. z  z  =  x  ->  ( A. z ph  <->  A. x ph ) )
54biimprd 158 . 2  |-  ( A. z  z  =  x  ->  ( A. x ph  ->  A. z ph )
)
61, 2, 5sylsyld 58 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557
This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786
This theorem is referenced by:  a16gb  1888  a16nf  1889
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