Theorem bj-axd2d 32577

Description: This implication, proved using only ax-gen 1722 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 32576. (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

Step	Hyp	Ref	Expression
1		pm2.27 42	. 2 |- ( A. x T. -> ( ( A. x T. -> E. x T. ) -> E. x T. ) )
2		tru 1487	. 2 |- T.
3	1, 2	mpg 1724	1 |- ( ( A. x T. -> E. x T. ) -> E. x T. )

Syntax hints:   -> wi 4   A.wal 1481   T. wtru 1484   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722

This theorem depends on definitions:  df-bi 197  df-tru 1486

This theorem is referenced by: (None)