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Theorem ceqsex2v 3098
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 1728 . 2  |-  F/ x ps
2 nfv 1728 . 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 3097 1  |-  ( E. x E. y ( x  =  A  /\  y  =  B  /\  ph )  <->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 974    = wceq 1405   E.wex 1633    e. wcel 1842   _Vcvv 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-v 3061
This theorem is referenced by:  ceqsex3v  3099  ceqsex4v  3100  ispos  15900  elfuns  30253  brimg  30275  brapply  30276  brsuccf  30279  brrestrict  30287  dfrdg4  30289  diblsmopel  34191
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