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Theorem bj-axd2d 32431
Description: This implication, proved using only ax-gen 1592 on top of propositional calculus (hence holding, up to the standard interpretation, in any modal logic), shows that the axiom scheme  |-  ( A. x ph  ->  E. x ph ) implies the axiom scheme  |-  E. x T.. These correspond to the modal axiom (D), and in predicate calculus, they assert that the universe of discourse is nonempty. For the converse, see bj-axdd2 32430. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-axd2d  |-  ( ( A. x T.  ->  E. x T.  )  ->  E. x T.  )

Proof of Theorem bj-axd2d
StepHypRef Expression
1 pm2.27 39 . 2  |-  ( A. x T.  ->  ( ( A. x T.  ->  E. x T.  )  ->  E. x T.  )
)
2 tru 1374 . 2  |- T.
31, 2mpg 1594 1  |-  ( ( A. x T.  ->  E. x T.  )  ->  E. x T.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1368   T. wtru 1371   E.wex 1587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592
This theorem depends on definitions:  df-bi 185  df-tru 1373
This theorem is referenced by: (None)
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