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Theorem ceqsal 2813
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
Hypotheses
Ref Expression
ceqsal.1  |-  F/ x ps
ceqsal.2  |-  A  e. 
_V
ceqsal.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ceqsal  |-  ( A. x ( x  =  A  ->  ph )  <->  ps )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem ceqsal
StepHypRef Expression
1 ceqsal.2 . 2  |-  A  e. 
_V
2 ceqsal.1 . . 3  |-  F/ x ps
3 ceqsal.3 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
42, 3ceqsalg 2812 . 2  |-  ( A  e.  _V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
51, 4ax-mp 8 1  |-  ( A. x ( x  =  A  ->  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   F/wnf 1531    = wceq 1623    e. wcel 1684   _Vcvv 2788
This theorem is referenced by:  ceqsalv  2814  aomclem6  27156
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-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790
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