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Theorem rspec2 2655
Description: Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)
Hypothesis
Ref Expression
rspec2.1  |-  A. x  e.  A  A. y  e.  B  ph
Assertion
Ref Expression
rspec2  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ph )

Proof of Theorem rspec2
StepHypRef Expression
1 rspec2.1 . . 3  |-  A. x  e.  A  A. y  e.  B  ph
21rspec 2620 . 2  |-  ( x  e.  A  ->  A. y  e.  B  ph )
32r19.21bi 2654 1  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   A.wral 2556
This theorem is referenced by:  rspec3  2656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ral 2561
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