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Theorem ceqsralv 2849
Description: Restricted quantifier version of ceqsalv 2848. (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 1610 . 2  |-  F/ x ps
2 ceqsralv.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32ax-gen 1537 . 2  |-  A. x
( x  =  A  ->  ( ph  <->  ps )
)
4 ceqsralt 2845 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  B )  ->  ( A. x  e.  B  ( x  =  A  ->  ph )  <->  ps )
)
51, 3, 4mp3an12 1267 1  |-  ( A  e.  B  ->  ( A. x  e.  B  ( x  =  A  ->  ph )  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1531   F/wnf 1535    = wceq 1633    e. wcel 1701   A.wral 2577
This theorem is referenced by:  eqreu  2991  sqr2irr  12574  acsfn  13610  ovolgelb  18892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-11 1732  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-ral 2582  df-v 2824
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