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Theorem rgen2 2518
Description: Generalization rule for restricted quantification. (Contributed by NM, 30-May-1999.)
Hypothesis
Ref Expression
rgen2.1  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ph )
Assertion
Ref Expression
rgen2  |-  A. x  e.  A  A. y  e.  B  ph
Distinct variable groups:    x, y    y, A
Allowed substitution hints:    ph( x, y)    A( x)    B( x, y)

Proof of Theorem rgen2
StepHypRef Expression
1 rgen2.1 . . 3  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ph )
21ralrimiva 2505 . 2  |-  ( x  e.  A  ->  A. y  e.  B  ph )
32rgen 2485 1  |-  A. x  e.  A  A. y  e.  B  ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1480   A.wral 2416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-4 1487  ax-17 1506
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-ral 2421
This theorem is referenced by:  rgen3  2519  f1stres  6057  f2ndres  6058  exmidonfinlem  7049  divfnzn  9420  txuni2  12435  divcnap  12734  abscncf  12751  recncf  12752  imcncf  12753  cjcncf  12754  ioocosf1o  12948
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