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Theorem ceqsex2v 2753
Description: Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
Hypotheses
Ref Expression
ceqsex2v.1  |-  A  e. 
_V
ceqsex2v.2  |-  B  e. 
_V
ceqsex2v.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
ceqsex2v.4  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
ceqsex2v  |-  ( E. x E. y ( x  =  A  /\  y  =  B  /\  ph )  <->  ch )
Distinct variable groups:    x, y, A   
x, B, y    ps, x    ch, y
Allowed substitution hints:    ph( x, y)    ps( y)    ch( x)

Proof of Theorem ceqsex2v
StepHypRef Expression
1 nfv 1508 . 2  |-  F/ x ps
2 nfv 1508 . 2  |-  F/ y ch
3 ceqsex2v.1 . 2  |-  A  e. 
_V
4 ceqsex2v.2 . 2  |-  B  e. 
_V
5 ceqsex2v.3 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
6 ceqsex2v.4 . 2  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
71, 2, 3, 4, 5, 6ceqsex2 2752 1  |-  ( E. x E. y ( x  =  A  /\  y  =  B  /\  ph )  <->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    /\ w3a 963    = wceq 1335   E.wex 1472    e. wcel 2128   _Vcvv 2712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714
This theorem is referenced by:  ceqsex3v  2754  ceqsex4v  2755
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