ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  19.37-1 Unicode version

Theorem 19.37-1 1637
Description: One direction of Theorem 19.37 of [Margaris] p. 90. The converse holds in classical logic but not, in general, here. (Contributed by Jim Kingdon, 21-Jun-2018.)
Hypothesis
Ref Expression
19.37-1.1  |-  F/ x ph
Assertion
Ref Expression
19.37-1  |-  ( E. x ( ph  ->  ps )  ->  ( ph  ->  E. x ps )
)

Proof of Theorem 19.37-1
StepHypRef Expression
1 19.37-1.1 . . 3  |-  F/ x ph
2119.3 1518 . 2  |-  ( A. x ph  <->  ph )
3 19.35-1 1588 . 2  |-  ( E. x ( ph  ->  ps )  ->  ( A. x ph  ->  E. x ps ) )
42, 3syl5bir 152 1  |-  ( E. x ( ph  ->  ps )  ->  ( ph  ->  E. x ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1314   F/wnf 1421   E.wex 1453
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 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-nf 1422
This theorem is referenced by:  19.37aiv  1638  spcimegft  2738  eqvincg  2783
  Copyright terms: Public domain W3C validator