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Theorem ceqsex 2835
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 470 . . 3  |-  ( ( x  =  A  /\  ph )  ->  ps )
41, 3exlimi 1813 . 2  |-  ( E. x ( x  =  A  /\  ph )  ->  ps )
52biimprcd 216 . . . 4  |-  ( ps 
->  ( x  =  A  ->  ph ) )
61, 5alrimi 1757 . . 3  |-  ( ps 
->  A. x ( x  =  A  ->  ph )
)
7 ceqsex.2 . . . 4  |-  A  e. 
_V
87isseti 2807 . . 3  |-  E. x  x  =  A
9 exintr 1604 . . 3  |-  ( A. x ( x  =  A  ->  ph )  -> 
( E. x  x  =  A  ->  E. x
( x  =  A  /\  ph ) ) )
106, 8, 9ee10 1366 . 2  |-  ( ps 
->  E. x ( x  =  A  /\  ph ) )
114, 10impbii 180 1  |-  ( E. x ( x  =  A  /\  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531   F/wnf 1534    = wceq 1632    e. wcel 1696   _Vcvv 2801
This theorem is referenced by:  ceqsexv  2836  ceqsex2  2837  dprd2d2  15295
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
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