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Theorem ceqsrex2v 2903
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 630 . . . . . 6  |-  ( ( ( x  =  A  /\  y  =  B )  /\  ph )  <->  ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
21rexbii 2568 . . . . 5  |-  ( E. y  e.  D  ( ( x  =  A  /\  y  =  B )  /\  ph )  <->  E. y  e.  D  ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
3 r19.42v 2694 . . . . 5  |-  ( E. y  e.  D  ( x  =  A  /\  ( y  =  B  /\  ph ) )  <-> 
( x  =  A  /\  E. y  e.  D  ( y  =  B  /\  ph )
) )
42, 3bitri 240 . . . 4  |-  ( E. y  e.  D  ( ( x  =  A  /\  y  =  B )  /\  ph )  <->  ( x  =  A  /\  E. y  e.  D  ( y  =  B  /\  ph ) ) )
54rexbii 2568 . . 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 684 . . . . 5  |-  ( x  =  A  ->  (
( y  =  B  /\  ph )  <->  ( y  =  B  /\  ps )
) )
87rexbidv 2564 . . . 4  |-  ( x  =  A  ->  ( E. y  e.  D  ( y  =  B  /\  ph )  <->  E. y  e.  D  ( y  =  B  /\  ps )
) )
98ceqsrexv 2901 . . 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 248 . 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 2901 . 2  |-  ( B  e.  D  ->  ( E. y  e.  D  ( y  =  B  /\  ps )  <->  ch )
)
1310, 12sylan9bb 680 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544
This theorem is referenced by:  opiota  6290  brdom7disj  8156  brdom6disj  8157  lsmspsn  15837
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-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-v 2790
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