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Theorem rexalim 2402
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 2398 . . 3  |-  ( A. x  e.  A  -.  ph  <->  -. 
E. x  e.  A  ph )
21biimpi 119 . 2  |-  ( A. x  e.  A  -.  ph 
->  -.  E. x  e.  A  ph )
32con2i 599 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 2388   E.wrex 2389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-5 1404  ax-gen 1406  ax-ie2 1451
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-fal 1318  df-ral 2393  df-rex 2394
This theorem is referenced by:  infnlbti  6863
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