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Theorem rsp2 2831
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 2823 . . 3  |-  ( A. x  e.  A  A. y  e.  B  ph  ->  ( x  e.  A  ->  A. y  e.  B  ph ) )
2 rsp 2823 . . 3  |-  ( A. y  e.  B  ph  ->  ( y  e.  B  ->  ph ) )
31, 2syl6 33 . 2  |-  ( A. x  e.  A  A. y  e.  B  ph  ->  ( x  e.  A  -> 
( y  e.  B  ->  ph ) ) )
43impd 431 1  |-  ( A. x  e.  A  A. y  e.  B  ph  ->  ( ( x  e.  A  /\  y  e.  B
)  ->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1762   A.wral 2807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-12 1798
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592  df-ral 2812
This theorem is referenced by:  ralcom2  3019  disjxiun  4437  solin  4816  mpt2curryd  6988  cmncom  16603  cnmpt21  19900  cnmpt2t  19902  cnmpt22  19903  cnmptcom  19907  frgrawopreglem5  24711  subgoablo  24975  htthlem  25496  prtlem14  30206  islptre  31116
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