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Theorem ceqsex2 3125
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 986 . . . . 5  |-  ( ( x  =  A  /\  y  =  B  /\  ph )  <->  ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
21exbii 1714 . . . 4  |-  ( E. y ( x  =  A  /\  y  =  B  /\  ph )  <->  E. y ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
3 19.42v 1826 . . . 4  |-  ( E. y ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  ( x  =  A  /\  E. y
( y  =  B  /\  ph ) ) )
42, 3bitri 252 . . 3  |-  ( E. y ( x  =  A  /\  y  =  B  /\  ph )  <->  ( x  =  A  /\  E. y ( y  =  B  /\  ph )
) )
54exbii 1714 . 2  |-  ( E. x E. y ( x  =  A  /\  y  =  B  /\  ph )  <->  E. x ( x  =  A  /\  E. y ( y  =  B  /\  ph )
) )
6 nfv 1754 . . . . 5  |-  F/ x  y  =  B
7 ceqsex2.1 . . . . 5  |-  F/ x ps
86, 7nfan 1986 . . . 4  |-  F/ x
( y  =  B  /\  ps )
98nfex 2006 . . 3  |-  F/ x E. y ( y  =  B  /\  ps )
10 ceqsex2.3 . . 3  |-  A  e. 
_V
11 ceqsex2.5 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
1211anbi2d 708 . . . 4  |-  ( x  =  A  ->  (
( y  =  B  /\  ph )  <->  ( y  =  B  /\  ps )
) )
1312exbidv 1761 . . 3  |-  ( x  =  A  ->  ( E. y ( y  =  B  /\  ph )  <->  E. y ( y  =  B  /\  ps )
) )
149, 10, 13ceqsex 3123 . 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 3123 . 2  |-  ( E. y ( y  =  B  /\  ps )  <->  ch )
195, 14, 183bitri 274 1  |-  ( E. x E. y ( x  =  A  /\  y  =  B  /\  ph )  <->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1659   F/wnf 1663    e. wcel 1870   _Vcvv 3087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-v 3089
This theorem is referenced by:  ceqsex2v  3126
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