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Theorem ralrimivv 2454
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 24-Jul-2004.)
Hypothesis
Ref Expression
ralrimivv.1  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ps ) )
Assertion
Ref Expression
ralrimivv  |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ps )
Distinct variable groups:    x, y, ph    y, A
Allowed substitution hints:    ps( x, y)    A( x)    B( x, y)

Proof of Theorem ralrimivv
StepHypRef Expression
1 ralrimivv.1 . . . 4  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ps ) )
21expd 254 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ( y  e.  B  ->  ps ) ) )
32ralrimdv 2452 . 2  |-  ( ph  ->  ( x  e.  A  ->  A. y  e.  B  ps ) )
43ralrimiv 2445 1  |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1438   A.wral 2359
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 1381  ax-gen 1383  ax-4 1445  ax-17 1464
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-ral 2364
This theorem is referenced by:  ralrimivva  2455  ralrimdvv  2457  reuind  2818  ssrel2  4516  f1o2ndf1  5975  smoiso  6049  nndifsnid  6246  receuap  8112  lbreu  8378
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