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Theorem ceqsrex2v 3073
Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.)
Hypotheses
Ref Expression
ceqsrex2v.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
ceqsrex2v.2  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
ceqsrex2v  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( E. x  e.  C  E. y  e.  D  ( ( x  =  A  /\  y  =  B )  /\  ph ) 
<->  ch ) )
Distinct variable groups:    x, y, A    x, B, y    x, C    x, D, y    ps, x    ch, y
Allowed substitution hints:    ph( x, y)    ps( y)    ch( x)    C( y)

Proof of Theorem ceqsrex2v
StepHypRef Expression
1 anass 632 . . . . . 6  |-  ( ( ( x  =  A  /\  y  =  B )  /\  ph )  <->  ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
21rexbii 2732 . . . . 5  |-  ( E. y  e.  D  ( ( x  =  A  /\  y  =  B )  /\  ph )  <->  E. y  e.  D  ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
3 r19.42v 2864 . . . . 5  |-  ( E. y  e.  D  ( x  =  A  /\  ( y  =  B  /\  ph ) )  <-> 
( x  =  A  /\  E. y  e.  D  ( y  =  B  /\  ph )
) )
42, 3bitri 242 . . . 4  |-  ( E. y  e.  D  ( ( x  =  A  /\  y  =  B )  /\  ph )  <->  ( x  =  A  /\  E. y  e.  D  ( y  =  B  /\  ph ) ) )
54rexbii 2732 . . 3  |-  ( E. x  e.  C  E. y  e.  D  (
( x  =  A  /\  y  =  B )  /\  ph )  <->  E. x  e.  C  ( x  =  A  /\  E. y  e.  D  ( y  =  B  /\  ph ) ) )
6 ceqsrex2v.1 . . . . . 6  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
76anbi2d 686 . . . . 5  |-  ( x  =  A  ->  (
( y  =  B  /\  ph )  <->  ( y  =  B  /\  ps )
) )
87rexbidv 2728 . . . 4  |-  ( x  =  A  ->  ( E. y  e.  D  ( y  =  B  /\  ph )  <->  E. y  e.  D  ( y  =  B  /\  ps )
) )
98ceqsrexv 3071 . . 3  |-  ( A  e.  C  ->  ( E. x  e.  C  ( x  =  A  /\  E. y  e.  D  ( y  =  B  /\  ph ) )  <->  E. y  e.  D  ( y  =  B  /\  ps ) ) )
105, 9syl5bb 250 . 2  |-  ( A  e.  C  ->  ( E. x  e.  C  E. y  e.  D  ( ( x  =  A  /\  y  =  B )  /\  ph ) 
<->  E. y  e.  D  ( y  =  B  /\  ps ) ) )
11 ceqsrex2v.2 . . 3  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
1211ceqsrexv 3071 . 2  |-  ( B  e.  D  ->  ( E. y  e.  D  ( y  =  B  /\  ps )  <->  ch )
)
1310, 12sylan9bb 682 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( E. x  e.  C  E. y  e.  D  ( ( x  =  A  /\  y  =  B )  /\  ph ) 
<->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708
This theorem is referenced by:  opiota  6537  brdom7disj  8411  brdom6disj  8412  lsmspsn  16158
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-v 2960
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