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Theorem r19.37 2581
Description: Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. In classical logic the converse would hold if  A has at least one element, but that is not sufficient in intuitionistic logic. (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypothesis
Ref Expression
r19.37.1  |-  F/ x ph
Assertion
Ref Expression
r19.37  |-  ( E. x  e.  A  (
ph  ->  ps )  -> 
( ph  ->  E. x  e.  A  ps )
)

Proof of Theorem r19.37
StepHypRef Expression
1 r19.37.1 . . 3  |-  F/ x ph
2 ax-1 6 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ph ) )
31, 2ralrimi 2501 . 2  |-  ( ph  ->  A. x  e.  A  ph )
4 r19.35-1 2579 . 2  |-  ( E. x  e.  A  (
ph  ->  ps )  -> 
( A. x  e.  A  ph  ->  E. x  e.  A  ps )
)
53, 4syl5 32 1  |-  ( E. x  e.  A  (
ph  ->  ps )  -> 
( ph  ->  E. x  e.  A  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1436    e. wcel 1480   A.wral 2414   E.wrex 2415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-ral 2419  df-rex 2420
This theorem is referenced by:  r19.37av  2582
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