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Theorem ceqsralv 3135
Description: Restricted quantifier version of ceqsalv 3134. (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 1712 . 2  |-  F/ x ps
2 ceqsralv.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32ax-gen 1623 . 2  |-  A. x
( x  =  A  ->  ( ph  <->  ps )
)
4 ceqsralt 3130 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) )  /\  A  e.  B )  ->  ( A. x  e.  B  ( x  =  A  ->  ph )  <->  ps )
)
51, 3, 4mp3an12 1312 1  |-  ( A  e.  B  ->  ( A. x  e.  B  ( x  =  A  ->  ph )  <->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1396    = wceq 1398   F/wnf 1621    e. wcel 1823   A.wral 2804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-ral 2809  df-v 3108
This theorem is referenced by:  eqreu  3288  sqrt2irr  14066  acsfn  15148  ovolgelb  22057
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