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Theorem eximal 1677
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 alimex 1714. (Contributed by BJ, 12-May-2019.)
Assertion
Ref Expression
eximal  |-  ( ( E. x ph  ->  ps )  <->  ( -.  ps  ->  A. x  -.  ph ) )

Proof of Theorem eximal
StepHypRef Expression
1 df-ex 1675 . . 3  |-  ( E. x ph  <->  -.  A. x  -.  ph )
21imbi1i 331 . 2  |-  ( ( E. x ph  ->  ps )  <->  ( -.  A. x  -.  ph  ->  ps )
)
3 con1b 339 . 2  |-  ( ( -.  A. x  -.  ph 
->  ps )  <->  ( -.  ps  ->  A. x  -.  ph ) )
42, 3bitri 257 1  |-  ( ( E. x ph  ->  ps )  <->  ( -.  ps  ->  A. x  -.  ph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189   A.wal 1453   E.wex 1674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 190  df-ex 1675
This theorem is referenced by:  ax5e  1771  19.23t  2002  xfree2  28154  bj-nalnalimiOLD  31269  bj-exalimi  31271
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