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Theorem equsalh 1901
Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
equsalh.1  |-  ( ps 
->  A. x ps )
equsalh.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
equsalh  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )

Proof of Theorem equsalh
StepHypRef Expression
1 equsalh.1 . . 3  |-  ( ps 
->  A. x ps )
21nfi 1538 . 2  |-  F/ x ps
3 equsalh.2 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
42, 3equsal 1900 1  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527
This theorem is referenced by:  dvelimh  1904  dvelimALT  2072  dvelimf-o  2119  dvelimfALT2OLD  28498
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532
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