Theorem bj-modal5e 32636

Description: Dual statement of hbe1 2021 (which is the real modal-5 2032). See also axc7 2132 and axc7e 2133. (Contributed by BJ, 21-Dec-2020.)

Assertion

Ref	Expression
bj-modal5e	|- ( E. x A. x ph -> A. x ph )

Proof of Theorem bj-modal5e

Step	Hyp	Ref	Expression
1		hbn1 2020	. . 3 |- ( -. A. x ph -> A. x -. A. x ph )
2		alnex 1706	. . 3 |- ( A. x -. A. x ph <-> -. E. x A. x ph )
3	1, 2	sylib 208	. 2 |- ( -. A. x ph -> -. E. x A. x ph )
4	3	con4i 113	1 |- ( E. x A. x ph -> A. 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  ax-10 2019

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

This theorem is referenced by:  bj-19.41al  32637  bj-sb56  32639