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Theorem ceqsexgv 3193
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.)
Hypothesis
Ref Expression
ceqsexgv.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ceqsexgv  |-  ( A  e.  V  ->  ( E. x ( x  =  A  /\  ph )  <->  ps ) )
Distinct variable groups:    x, A    ps, x
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem ceqsexgv
StepHypRef Expression
1 nfv 1674 . 2  |-  F/ x ps
2 ceqsexgv.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
31, 2ceqsexg 3192 1  |-  ( A  e.  V  ->  ( E. x ( x  =  A  /\  ph )  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-v 3074
This theorem is referenced by:  ceqsrexv  3194  clel3g  3198  elxp5  6628  xpsnen  7500  isssc  14847  metuel2  20281  isgrpo  23830  ismgm  23954  ceqsex3vOLD  29149  bj-finsumval0  32902  pmapjat1  33816
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