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Theorem ceqsex2v 2825
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 1605 . 2  |-  F/ x ps
2 nfv 1605 . 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 2824 1  |-  ( E. x E. y ( x  =  A  /\  y  =  B  /\  ph )  <->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788
This theorem is referenced by:  ceqsex3v  2826  ceqsex4v  2827  ispos  14081  brapply  24477  brsuccf  24480  brrestrict  24487  dfrdg4  24488  diblsmopel  31361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790
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