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Theorem 19.35-1 1558
Description: Forward direction of Theorem 19.35 of [Margaris] p. 90. The converse holds for classical logic but not (for all propositions) in intuitionistic logic (Contributed by Mario Carneiro, 2-Feb-2015.)
Assertion
Ref Expression
19.35-1  |-  ( E. x ( ph  ->  ps )  ->  ( A. x ph  ->  E. x ps ) )

Proof of Theorem 19.35-1
StepHypRef Expression
1 19.29 1554 . . 3  |-  ( ( A. x ph  /\  E. x ( ph  ->  ps ) )  ->  E. x
( ph  /\  ( ph  ->  ps ) ) )
2 pm3.35 339 . . . 4  |-  ( (
ph  /\  ( ph  ->  ps ) )  ->  ps )
32eximi 1534 . . 3  |-  ( E. x ( ph  /\  ( ph  ->  ps )
)  ->  E. x ps )
41, 3syl 14 . 2  |-  ( ( A. x ph  /\  E. x ( ph  ->  ps ) )  ->  E. x ps )
54expcom 114 1  |-  ( E. x ( ph  ->  ps )  ->  ( A. x ph  ->  E. x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1285   E.wex 1424
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-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-ial 1470
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  19.35i  1559  19.25  1560  19.36-1  1606  19.37-1  1607  spimt  1668  sbequi  1764
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