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Theorem ceqsal 2583
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 2582 . 2  |-  ( A  e.  _V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
51, 4ax-mp 7 1  |-  ( A. x ( x  =  A  ->  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98   A.wal 1241    = wceq 1243   F/wnf 1349    e. wcel 1393   _Vcvv 2557
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-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559
This theorem is referenced by:  ceqsalv  2584
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