Theorem bj-axtd 32578

Description: This implication, proved from propositional calculus only (hence holding, up to the standard interpretation, in any modal logic), shows that the axiom scheme |- ( A. x ph -> ph ) (modal T) implies the axiom scheme |- ( A. x ph -> E. x ph ) (modal D). See also bj-axdd2 32576 and bj-axd2d 32577. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem bj-axtd

Step	Hyp	Ref	Expression
1		con2 130	. . 3 |- ( ( A. x -. ph -> -. ph ) -> ( ph -> -. A. x -. ph ) )
2		df-ex 1705	. . 3 |- ( E. x ph <-> -. A. x -. ph )
3	1, 2	syl6ibr 242	. 2 |- ( ( A. x -. ph -> -. ph ) -> ( ph -> E. x ph ) )
4	3	imim2d 57	1 |- ( ( A. x -. ph -> -. ph ) -> ( ( A. x ph -> ph ) -> ( A. x ph -> E. x ph ) ) )

Syntax hints:   -. wn 3   -> 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

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

This theorem is referenced by: (None)