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Theorem rsp2 2760
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 2758 . . 3  |-  ( A. x  e.  A  A. y  e.  B  ph  ->  ( x  e.  A  ->  A. y  e.  B  ph ) )
2 rsp 2758 . . 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 1725   A.wral 2697
This theorem is referenced by:  ralcom2  2864  disjxiun  4201  solin  4518  cmncom  15420  cnmpt21  17695  cnmpt2t  17697  cnmpt22  17698  cnmptcom  17702  subgoablo  21891  htthlem  22412  prtlem14  26714  frgrawopreglem5  28374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-ral 2702
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