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Theorem rsp2 2711
Description: Restricted specialization. (Contributed by NM, 11-Feb-1997.)
Assertion
Ref Expression
rsp2  |-  ( A. x  e.  A  A. y  e.  B  ph  ->  ( ( x  e.  A  /\  y  e.  B
)  ->  ph ) )

Proof of Theorem rsp2
StepHypRef Expression
1 rsp 2709 . . 3  |-  ( A. x  e.  A  A. y  e.  B  ph  ->  ( x  e.  A  ->  A. y  e.  B  ph ) )
2 rsp 2709 . . 3  |-  ( A. y  e.  B  ph  ->  ( y  e.  B  ->  ph ) )
31, 2syl6 31 . 2  |-  ( A. x  e.  A  A. y  e.  B  ph  ->  ( x  e.  A  -> 
( y  e.  B  ->  ph ) ) )
43imp3a 421 1  |-  ( A. x  e.  A  A. y  e.  B  ph  ->  ( ( x  e.  A  /\  y  e.  B
)  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717   A.wral 2649
This theorem is referenced by:  ralcom2  2815  disjxiun  4150  solin  4467  cmncom  15355  cnmpt21  17624  cnmpt2t  17626  cnmpt22  17627  cnmptcom  17631  subgoablo  21747  htthlem  22268  prtlem14  26414  frgrawopreglem5  27800
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-11 1753
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-ral 2654
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