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Theorem ceqsex3OLD 26829
Description: Version of ceqsex 2835 with an antecedent instead of a hypothesis. (Use ceqsexg 2912 instead of this one. --NM 13-Aug-11) (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ceqsex3.1OLD  |-  ( ps 
->  A. x ps )
ceqsex3.2OLD  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ceqsex3OLD  |-  ( A  e.  _V  ->  ( E. x ( x  =  A  /\  ph )  <->  ps ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem ceqsex3OLD
StepHypRef Expression
1 ceqsex3.1OLD . . 3  |-  ( ps 
->  A. x ps )
2 ceqsex3.2OLD . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32biimpa 470 . . 3  |-  ( ( x  =  A  /\  ph )  ->  ps )
41, 3exlimih 1741 . 2  |-  ( E. x ( x  =  A  /\  ph )  ->  ps )
5 isset 2805 . . . 4  |-  ( A  e.  _V  <->  E. x  x  =  A )
62biimprcd 216 . . . . . 6  |-  ( ps 
->  ( x  =  A  ->  ph ) )
71, 6alrimih 1555 . . . . 5  |-  ( ps 
->  A. x ( x  =  A  ->  ph )
)
8 exintr 1604 . . . . 5  |-  ( A. x ( x  =  A  ->  ph )  -> 
( E. x  x  =  A  ->  E. x
( x  =  A  /\  ph ) ) )
97, 8syl 15 . . . 4  |-  ( ps 
->  ( E. x  x  =  A  ->  E. x
( x  =  A  /\  ph ) ) )
105, 9syl5bi 208 . . 3  |-  ( ps 
->  ( A  e.  _V  ->  E. x ( x  =  A  /\  ph ) ) )
1110com12 27 . 2  |-  ( A  e.  _V  ->  ( ps  ->  E. x ( x  =  A  /\  ph ) ) )
124, 11impbid2 195 1  |-  ( A  e.  _V  ->  ( 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    = wceq 1632    e. wcel 1696   _Vcvv 2801
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-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
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