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Theorem bj-nalnaleximiOLD 31001
Description: An inference for distributing quantifiers over a double implication. The general statement that speimfw 1785 proves. (Contributed by BJ, 12-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-nalnaleximiOLD.1  |-  ( ch 
->  ( ph  ->  ps ) )
Assertion
Ref Expression
bj-nalnaleximiOLD  |-  ( -. 
A. x  -.  ch  ->  ( A. x ph  ->  E. x ps )
)

Proof of Theorem bj-nalnaleximiOLD
StepHypRef Expression
1 bj-nalnaleximiOLD.1 . . 3  |-  ( ch 
->  ( ph  ->  ps ) )
21eximi 1703 . 2  |-  ( E. x ch  ->  E. x
( ph  ->  ps )
)
3 df-ex 1660 . 2  |-  ( E. x ch  <->  -.  A. x  -.  ch )
4 19.35 1734 . 2  |-  ( E. x ( ph  ->  ps )  <->  ( A. x ph  ->  E. x ps )
)
52, 3, 43imtr3i 268 1  |-  ( -. 
A. x  -.  ch  ->  ( A. x ph  ->  E. x ps )
)
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:  bj-nalnalimiOLD  31002
  Copyright terms: Public domain W3C validator