Theorem bj-axdd2 32576

Description: This implication, proved using only ax-gen 1722 and ax-4 1737 on top of propositional calculus (hence holding, up to the standard interpretation, in any normal modal logic), shows that the axiom scheme |- E. x T. implies the axiom scheme |- ( A. x ph -> E. x 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-axd2d 32577. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-axdd2	|- ( E. x ph -> ( A. x ps -> E. x ps ) )

Proof of Theorem bj-axdd2

Step	Hyp	Ref	Expression
1		ala1 1741	. . 3 |- ( A. x ps -> A. x ( ph -> ps ) )
2		exim 1761	. . 3 |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) )
3	1, 2	syl 17	. 2 |- ( A. x ps -> ( E. x ph -> E. x ps ) )
4	3	com12 32	1 |- ( E. x ph -> ( A. x ps -> E. x ps ) )

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

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by: (None)