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Theorem ceqsex2 2824
Description: Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
Hypotheses
Ref Expression
ceqsex2.1  |-  F/ x ps
ceqsex2.2  |-  F/ y ch
ceqsex2.3  |-  A  e. 
_V
ceqsex2.4  |-  B  e. 
_V
ceqsex2.5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
ceqsex2.6  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
ceqsex2  |-  ( E. x E. y ( x  =  A  /\  y  =  B  /\  ph )  <->  ch )
Distinct variable groups:    x, y, A    x, B, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    ch( x, y)

Proof of Theorem ceqsex2
StepHypRef Expression
1 3anass 938 . . . . 5  |-  ( ( x  =  A  /\  y  =  B  /\  ph )  <->  ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
21exbii 1569 . . . 4  |-  ( E. y ( x  =  A  /\  y  =  B  /\  ph )  <->  E. y ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
3 19.42v 1846 . . . 4  |-  ( E. y ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  ( x  =  A  /\  E. y
( y  =  B  /\  ph ) ) )
42, 3bitri 240 . . 3  |-  ( E. y ( x  =  A  /\  y  =  B  /\  ph )  <->  ( x  =  A  /\  E. y ( y  =  B  /\  ph )
) )
54exbii 1569 . 2  |-  ( E. x E. y ( x  =  A  /\  y  =  B  /\  ph )  <->  E. x ( x  =  A  /\  E. y ( y  =  B  /\  ph )
) )
6 nfv 1605 . . . . 5  |-  F/ x  y  =  B
7 ceqsex2.1 . . . . 5  |-  F/ x ps
86, 7nfan 1771 . . . 4  |-  F/ x
( y  =  B  /\  ps )
98nfex 1767 . . 3  |-  F/ x E. y ( y  =  B  /\  ps )
10 ceqsex2.3 . . 3  |-  A  e. 
_V
11 ceqsex2.5 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
1211anbi2d 684 . . . 4  |-  ( x  =  A  ->  (
( y  =  B  /\  ph )  <->  ( y  =  B  /\  ps )
) )
1312exbidv 1612 . . 3  |-  ( x  =  A  ->  ( E. y ( y  =  B  /\  ph )  <->  E. y ( y  =  B  /\  ps )
) )
149, 10, 13ceqsex 2822 . 2  |-  ( E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) )  <->  E. y ( y  =  B  /\  ps )
)
15 ceqsex2.2 . . 3  |-  F/ y ch
16 ceqsex2.4 . . 3  |-  B  e. 
_V
17 ceqsex2.6 . . 3  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
1815, 16, 17ceqsex 2822 . 2  |-  ( E. y ( y  =  B  /\  ps )  <->  ch )
195, 14, 183bitri 262 1  |-  ( E. x E. y ( x  =  A  /\  y  =  B  /\  ph )  <->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528   F/wnf 1531    = wceq 1623    e. wcel 1684   _Vcvv 2788
This theorem is referenced by:  ceqsex2v  2825
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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