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Theorem r19.40 2861
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 454 . . 3  |-  ( (
ph  /\  ps )  ->  ph )
21reximi 2813 . 2  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  E. x  e.  A  ph )
3 simpr 458 . . 3  |-  ( (
ph  /\  ps )  ->  ps )
43reximi 2813 . 2  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  E. x  e.  A  ps )
52, 4jca 529 1  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  ( E. x  e.  A  ph  /\  E. x  e.  A  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   E.wrex 2706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1590  df-ral 2710  df-rex 2711
This theorem is referenced by:  rexanuz  12816  txflf  19420  metequiv2  19926  mzpcompact2lem  28930
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