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Theorem ceqsex 2790
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)
Hypotheses
Ref Expression
ceqsex.1  |-  F/ x ps
ceqsex.2  |-  A  e. 
_V
ceqsex.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ceqsex  |-  ( E. x ( x  =  A  /\  ph )  <->  ps )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem ceqsex
StepHypRef Expression
1 ceqsex.1 . . 3  |-  F/ x ps
2 ceqsex.3 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32biimpa 296 . . 3  |-  ( ( x  =  A  /\  ph )  ->  ps )
41, 3exlimi 1605 . 2  |-  ( E. x ( x  =  A  /\  ph )  ->  ps )
52biimprcd 160 . . . 4  |-  ( ps 
->  ( x  =  A  ->  ph ) )
61, 5alrimi 1533 . . 3  |-  ( ps 
->  A. x ( x  =  A  ->  ph )
)
7 ceqsex.2 . . . 4  |-  A  e. 
_V
87isseti 2760 . . 3  |-  E. x  x  =  A
9 exintr 1645 . . 3  |-  ( A. x ( x  =  A  ->  ph )  -> 
( E. x  x  =  A  ->  E. x
( x  =  A  /\  ph ) ) )
106, 8, 9mpisyl 1457 . 2  |-  ( ps 
->  E. x ( x  =  A  /\  ph ) )
114, 10impbii 126 1  |-  ( E. x ( x  =  A  /\  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1362    = wceq 1364   F/wnf 1471   E.wex 1503    e. wcel 2160   _Vcvv 2752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-v 2754
This theorem is referenced by:  ceqsexv  2791  ceqsex2  2792
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