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Theorem rexalim 2369
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 2365 . . 3  |-  ( A. x  e.  A  -.  ph  <->  -. 
E. x  e.  A  ph )
21biimpi 118 . 2  |-  ( A. x  e.  A  -.  ph 
->  -.  E. x  e.  A  ph )
32con2i 590 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 2355   E.wrex 2356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1379  ax-gen 1381  ax-ie2 1426
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-ral 2360  df-rex 2361
This theorem is referenced by:  infnlbti  6658
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