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Theorem ceqsralv 2644
Description: Restricted quantifier version of ceqsalv 2643. (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 1464 . 2  |-  F/ x ps
2 ceqsralv.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32ax-gen 1381 . 2  |-  A. x
( x  =  A  ->  ( ph  <->  ps )
)
4 ceqsralt 2640 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  B )  ->  ( A. x  e.  B  ( x  =  A  ->  ph )  <->  ps )
)
51, 3, 4mp3an12 1261 1  |-  ( A  e.  B  ->  ( A. x  e.  B  ( x  =  A  ->  ph )  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1285    = wceq 1287   F/wnf 1392    e. wcel 1436   A.wral 2355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-ral 2360  df-v 2617
This theorem is referenced by:  eqreu  2798  sqrt2irr  11016
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