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Theorem r19.37 2507
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 5 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ph ) )
31, 2ralrimi 2433 . 2  |-  ( ph  ->  A. x  e.  A  ph )
4 r19.35-1 2505 . 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 1390    e. wcel 1434   A.wral 2349   E.wrex 2350
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-ral 2354  df-rex 2355
This theorem is referenced by:  r19.37av  2508
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