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Theorem i19.24 1546
Description: Theorem 19.24 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1531, 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
i19.24  |-  ( ( A. x ph  ->  A. x ps )  ->  E. x ( ph  ->  ps ) )

Proof of Theorem i19.24
StepHypRef Expression
1 19.2 1545 . . 3  |-  ( A. x ps  ->  E. x ps )
21imim2i 12 . 2  |-  ( ( A. x ph  ->  A. 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  |-  ( ( A. x ph  ->  A. x ps )  ->  E. x ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257   E.wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416
This theorem depends on definitions:  df-bi 114
This theorem is referenced by: (None)
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