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Theorem ceqsralv 2985
Description: Restricted quantifier version of ceqsalv 2984. (Contributed by NM, 21-Jun-2013.)
Hypothesis
Ref Expression
ceqsralv.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ceqsralv  |-  ( A  e.  B  ->  ( A. x  e.  B  ( x  =  A  ->  ph )  <->  ps )
)
Distinct variable groups:    x, A    x, B    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem ceqsralv
StepHypRef Expression
1 nfv 1630 . 2  |-  F/ x ps
2 ceqsralv.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32ax-gen 1556 . 2  |-  A. x
( x  =  A  ->  ( ph  <->  ps )
)
4 ceqsralt 2981 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  B )  ->  ( A. x  e.  B  ( x  =  A  ->  ph )  <->  ps )
)
51, 3, 4mp3an12 1270 1  |-  ( A  e.  B  ->  ( A. x  e.  B  ( x  =  A  ->  ph )  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550   F/wnf 1554    = wceq 1653    e. wcel 1726   A.wral 2707
This theorem is referenced by:  eqreu  3128  sqr2irr  12850  acsfn  13886  ovolgelb  19378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-11 1762  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-3an 939  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-ral 2712  df-v 2960
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