Theorem bj-axd2d

Description: This implication, proved using only ax-gen 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. (substitute T. for ph ). 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 . (Contributed by BJ, 16-May-2019) Generalize from its instance with T. substituted for ph . (Revised by BJ, 20-Mar-2022) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	bj-axd2d	|- ( ( A. x T. -> E. x ph ) -> E. x ph )

Proof

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