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Theorem r19.40 2562
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 108 . . 3  |-  ( (
ph  /\  ps )  ->  ph )
21reximi 2506 . 2  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  E. x  e.  A  ph )
3 simpr 109 . . 3  |-  ( (
ph  /\  ps )  ->  ps )
43reximi 2506 . 2  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  E. x  e.  A  ps )
52, 4jca 304 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 103   E.wrex 2394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-ral 2398  df-rex 2399
This theorem is referenced by:  rexanuz  10728  metequiv2  12592
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