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Theorem i19.39 1572
Description: Theorem 19.39 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1556, and is a theorem of classical logic, but in intuitionistic logic it will only be provable for some propositions. (Contributed by Jim Kingdon, 22-Jul-2018.)
Hypothesis
Ref Expression
i19.24.1  |-  ( ( A. x ph  ->  E. x ps )  ->  E. x ( ph  ->  ps ) )
Assertion
Ref Expression
i19.39  |-  ( ( E. x ph  ->  E. x ps )  ->  E. x ( ph  ->  ps ) )

Proof of Theorem i19.39
StepHypRef Expression
1 19.2 1570 . . 3  |-  ( A. x ph  ->  E. x ph )
21imim1i 59 . 2  |-  ( ( E. x ph  ->  E. x ps )  -> 
( A. x ph  ->  E. x ps )
)
3 i19.24.1 . 2  |-  ( ( A. x ph  ->  E. x ps )  ->  E. x ( ph  ->  ps ) )
42, 3syl 14 1  |-  ( ( E. x ph  ->  E. x ps )  ->  E. x ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1283   E.wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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