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Theorem bj-exaleximi 31003
Description: An inference for distributing quantifiers over a double implication. (Almost) the general statement that speimfw 1785 proves. (Contributed by BJ, 29-Sep-2019.)
Hypothesis
Ref Expression
bj-exaleximi.1  |-  ( ph  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
bj-exaleximi  |-  ( E. x ph  ->  ( A. x ps  ->  E. x ch ) )

Proof of Theorem bj-exaleximi
StepHypRef Expression
1 bj-exaleximi.1 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
21eximi 1703 . 2  |-  ( E. x ph  ->  E. x
( ps  ->  ch ) )
3 19.35 1734 . 2  |-  ( E. x ( ps  ->  ch )  <->  ( A. x ps  ->  E. x ch )
)
42, 3sylib 199 1  |-  ( E. x ph  ->  ( A. x ps  ->  E. x ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> 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:  bj-exalimi  31004
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