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Theorem alimex 1700
Description: A utility theorem. An interesting case is when the same formula is substituted for both  ph and  ps, since then both implications express a type of non-freeness. See also eximal 1663. (Contributed by BJ, 12-May-2019.)
Assertion
Ref Expression
alimex  |-  ( (
ph  ->  A. x ps )  <->  ( E. x  -.  ps  ->  -.  ph ) )

Proof of Theorem alimex
StepHypRef Expression
1 alex 1695 . . 3  |-  ( A. x ps  <->  -.  E. x  -.  ps )
21imbi2i 314 . 2  |-  ( (
ph  ->  A. x ps )  <->  (
ph  ->  -.  E. x  -.  ps ) )
3 con2b 336 . 2  |-  ( (
ph  ->  -.  E. x  -.  ps )  <->  ( E. x  -.  ps  ->  -.  ph ) )
42, 3bitri 253 1  |-  ( (
ph  ->  A. x ps )  <->  ( E. x  -.  ps  ->  -.  ph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188   A.wal 1436   E.wex 1660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679
This theorem depends on definitions:  df-bi 189  df-ex 1661
This theorem is referenced by: (None)
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