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Theorem rgen2 2448
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 2435 . 2  |-  ( x  e.  A  ->  A. y  e.  B  ph )
32rgen 2417 1  |-  A. x  e.  A  A. y  e.  B  ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1434   A.wral 2349
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-4 1441  ax-17 1460
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-ral 2354
This theorem is referenced by:  rgen3  2449  f1stres  5817  f2ndres  5818  divfnzn  8787
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