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Theorem r19.40 2704
Description: Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
Assertion
Ref Expression
r19.40  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  ( E. x  e.  A  ph  /\  E. x  e.  A  ps ) )

Proof of Theorem r19.40
StepHypRef Expression
1 simpl 443 . . 3  |-  ( (
ph  /\  ps )  ->  ph )
21reximi 2663 . 2  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  E. x  e.  A  ph )
3 simpr 447 . . 3  |-  ( (
ph  /\  ps )  ->  ps )
43reximi 2663 . 2  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  E. x  e.  A  ps )
52, 4jca 518 1  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  ( E. x  e.  A  ph  /\  E. x  e.  A  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wrex 2557
This theorem is referenced by:  rexanuz  11845  txflf  17717  metequiv2  18072  mzpcompact2lem  26932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-ral 2561  df-rex 2562
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