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Theorem a16gb 2053
Description: A generalization of axiom ax-16 2223. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
a16gb  |-  ( A. x  x  =  y  ->  ( ph  <->  A. z ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem a16gb
StepHypRef Expression
1 a16g 2049 . 2  |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
2 sp 1764 . 2  |-  ( A. z ph  ->  ph )
31, 2impbid1 196 1  |-  ( A. x  x  =  y  ->  ( ph  <->  A. z ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550
This theorem is referenced by:  sbal  2206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555
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