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Theorem rexlimdvv 2484
Description: Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004.)
Hypothesis
Ref Expression
rexlimdvv.1  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ( ps  ->  ch ) ) )
Assertion
Ref Expression
rexlimdvv  |-  ( ph  ->  ( E. x  e.  A  E. y  e.  B  ps  ->  ch ) )
Distinct variable groups:    x, y, ph    ch, x, y    y, A
Allowed substitution hints:    ps( x, y)    A( x)    B( x, y)

Proof of Theorem rexlimdvv
StepHypRef Expression
1 rexlimdvv.1 . . . 4  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ( ps  ->  ch ) ) )
21expdimp 255 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
y  e.  B  -> 
( ps  ->  ch ) ) )
32rexlimdv 2477 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( E. y  e.  B  ps  ->  ch ) )
43rexlimdva 2478 1  |-  ( ph  ->  ( E. x  e.  A  E. y  e.  B  ps  ->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1434   E.wrex 2350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-ral 2354  df-rex 2355
This theorem is referenced by:  rexlimdvva  2485  f1oiso2  5497  xpdom2  6375  genpcdl  6771  genpcuu  6772  distrlem1prl  6834  distrlem1pru  6835  distrlem5prl  6838  distrlem5pru  6839  recexprlemss1l  6887  recexprlemss1u  6888  qaddcl  8801  qmulcl  8803  dvdsgcd  10545  gcddiv  10552
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