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Theorem 2r19.29 31841
Description: Double the quantifiers of theorem r19.29. (Contributed by Rodolfo Medina, 25-Sep-2010.)
Assertion
Ref Expression
2r19.29  |-  ( ( A. x  e.  A  A. y  e.  B  ph 
/\  E. x  e.  A  E. y  e.  B  ps )  ->  E. x  e.  A  E. y  e.  B  ( ph  /\ 
ps ) )

Proof of Theorem 2r19.29
StepHypRef Expression
1 r19.29 2939 . 2  |-  ( ( A. x  e.  A  A. y  e.  B  ph 
/\  E. x  e.  A  E. y  e.  B  ps )  ->  E. x  e.  A  ( A. y  e.  B  ph  /\  E. y  e.  B  ps ) )
2 r19.29 2939 . . 3  |-  ( ( A. y  e.  B  ph 
/\  E. y  e.  B  ps )  ->  E. y  e.  B  ( ph  /\ 
ps ) )
32reximi 2869 . 2  |-  ( E. x  e.  A  ( A. y  e.  B  ph 
/\  E. y  e.  B  ps )  ->  E. x  e.  A  E. y  e.  B  ( ph  /\ 
ps ) )
41, 3syl 17 1  |-  ( ( A. x  e.  A  A. y  e.  B  ph 
/\  E. x  e.  A  E. y  e.  B  ps )  ->  E. x  e.  A  E. y  e.  B  ( ph  /\ 
ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wral 2751   E.wrex 2752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1632  df-ral 2756  df-rex 2757
This theorem is referenced by:  prter2  31868
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