Theorem bj-modal4e 32705

Description: Dual statement of hba1 2151 (which is modal-4 ). (Contributed by BJ, 21-Dec-2020.)

Assertion

Ref	Expression
bj-modal4e	|- ( E. x E. x ph -> E. x ph )

Proof of Theorem bj-modal4e

Step	Hyp	Ref	Expression
1		hba1 2151	. . 3 |- ( A. x -. ph -> A. x A. x -. ph )
2		alnex 1706	. . 3 |- ( A. x -. ph <-> -. E. x ph )
3		2exnaln 1756	. . . 4 |- ( E. x E. x ph <-> -. A. x A. x -. ph )
4	3	con2bii 347	. . 3 |- ( A. x A. x -. ph <-> -. E. x E. x ph )
5	1, 2, 4	3imtr3i 280	. 2 |- ( -. E. x ph -> -. E. x E. x ph )
6	5	con4i 113	1 |- ( E. x E. 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  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)