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Theorem ceqsexv2d 23938
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.)
Hypotheses
Ref Expression
ceqsexv2d.1  |-  A  e. 
_V
ceqsexv2d.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
ceqsexv2d.3  |-  ps
Assertion
Ref Expression
ceqsexv2d  |-  E. x ph
Distinct variable groups:    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem ceqsexv2d
StepHypRef Expression
1 ceqsexv2d.3 . 2  |-  ps
2 ceqsexv2d.1 . . . 4  |-  A  e. 
_V
3 ceqsexv2d.2 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
42, 3ceqsexv 2951 . . 3  |-  ( E. x ( x  =  A  /\  ph )  <->  ps )
54biimpri 198 . 2  |-  ( ps 
->  E. x ( x  =  A  /\  ph ) )
6 simpr 448 . . 3  |-  ( ( x  =  A  /\  ph )  ->  ph )
76eximi 1582 . 2  |-  ( E. x ( x  =  A  /\  ph )  ->  E. x ph )
81, 5, 7mp2b 10 1  |-  E. x ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   _Vcvv 2916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-11 1757  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-v 2918
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