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Theorem 2r19.29 26703
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 2846 . 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 2846 . . 3  |-  ( ( A. y  e.  B  ph 
/\  E. y  e.  B  ps )  ->  E. y  e.  B  ( ph  /\ 
ps ) )
32reximi 2813 . 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 16 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 set class
Syntax hints:    -> wi 4    /\ wa 359   A.wral 2705   E.wrex 2706
This theorem is referenced by:  prter2  26730
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-ral 2710  df-rex 2711
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