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Theorem rexalim 2470
Description: Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
rexalim  |-  ( E. x  e.  A  ph  ->  -.  A. x  e.  A  -.  ph )

Proof of Theorem rexalim
StepHypRef Expression
1 ralnex 2465 . . 3  |-  ( A. x  e.  A  -.  ph  <->  -. 
E. x  e.  A  ph )
21biimpi 120 . 2  |-  ( A. x  e.  A  -.  ph 
->  -.  E. x  e.  A  ph )
32con2i 627 1  |-  ( E. x  e.  A  ph  ->  -.  A. x  e.  A  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wral 2455   E.wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie2 1494
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-ral 2460  df-rex 2461
This theorem is referenced by:  infnlbti  7024
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