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Theorem bj-exalimi 31004
Description: An inference for distributing quantifiers over a double implication. (Almost) the general statement that spimfw 1787 proves. (Contributed by BJ, 29-Sep-2019.)
Hypotheses
Ref Expression
bj-exalimi.1  |-  ( ph  ->  ( ps  ->  ch ) )
bj-exalimi.2  |-  ( E. x ph  ->  ( -.  ch  ->  A. x  -.  ch ) )
Assertion
Ref Expression
bj-exalimi  |-  ( E. x ph  ->  ( A. x ps  ->  ch ) )

Proof of Theorem bj-exalimi
StepHypRef Expression
1 bj-exalimi.1 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
21bj-exaleximi 31003 . 2  |-  ( E. x ph  ->  ( A. x ps  ->  E. x ch ) )
3 bj-exalimi.2 . . 3  |-  ( E. x ph  ->  ( -.  ch  ->  A. x  -.  ch ) )
4 eximal 1662 . . 3  |-  ( ( E. x ch  ->  ch )  <->  ( -.  ch  ->  A. x  -.  ch ) )
53, 4sylibr 215 . 2  |-  ( E. x ph  ->  ( E. x ch  ->  ch ) )
62, 5syld 45 1  |-  ( E. x ph  ->  ( A. x ps  ->  ch ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1435   E.wex 1659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678
This theorem depends on definitions:  df-bi 188  df-ex 1660
This theorem is referenced by: (None)
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