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Theorem rsp2 2775
Description: Restricted specialization, with two quantifiers. (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 2767 . . 3  |-  ( A. x  e.  A  A. y  e.  B  ph  ->  ( x  e.  A  ->  A. y  e.  B  ph ) )
2 rsp 2767 . . 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 ) ) )
43impd 429 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 367    e. wcel 1840   A.wral 2751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-12 1876
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1632  df-ral 2756
This theorem is referenced by:  ralcom2  2969  disjxiun  4389  solin  4764  mpt2curryd  6953  cmncom  17028  cnmpt21  20354  cnmpt2t  20356  cnmpt22  20357  cnmptcom  20361  frgrawopreglem5  25347  subgoablo  25608  htthlem  26129  prtlem14  31861  islptre  36960
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