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Theorem ceqsex 3070
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 482 . . 3  |-  ( ( x  =  A  /\  ph )  ->  ps )
41, 3exlimi 1920 . 2  |-  ( E. x ( x  =  A  /\  ph )  ->  ps )
52biimprcd 225 . . . 4  |-  ( ps 
->  ( x  =  A  ->  ph ) )
61, 5alrimi 1885 . . 3  |-  ( ps 
->  A. x ( x  =  A  ->  ph )
)
7 ceqsex.2 . . . 4  |-  A  e. 
_V
87isseti 3040 . . 3  |-  E. x  x  =  A
9 exintr 1710 . . 3  |-  ( A. x ( x  =  A  ->  ph )  -> 
( E. x  x  =  A  ->  E. x
( x  =  A  /\  ph ) ) )
106, 8, 9mpisyl 18 . 2  |-  ( ps 
->  E. x ( x  =  A  /\  ph ) )
114, 10impbii 188 1  |-  ( E. x ( x  =  A  /\  ph )  <->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1397    = wceq 1399   E.wex 1620   F/wnf 1624    e. wcel 1826   _Vcvv 3034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-12 1862  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-v 3036
This theorem is referenced by:  ceqsexv  3071  ceqsex2  3072
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