Theorem eximal 1707

Description: A utility theorem. An interesting case is when the same formula is substituted for both ph and ps, since then both implications express a type of non-freeness. See also alimex 1758. (Contributed by BJ, 12-May-2019.)

Assertion

Ref	Expression
eximal	|- ( ( E. x ph -> ps ) <-> ( -. ps -> A. x -. ph ) )

Proof of Theorem eximal

Step	Hyp	Ref	Expression
1		df-ex 1705	. . 3 |- ( E. x ph <-> -. A. x -. ph )
2	1	imbi1i 339	. 2 |- ( ( E. x ph -> ps ) <-> ( -. A. x -. ph -> ps ) )
3		con1b 348	. 2 |- ( ( -. A. x -. ph -> ps ) <-> ( -. ps -> A. x -. ph ) )
4	2, 3	bitri 264	1 |- ( ( E. x ph -> ps ) <-> ( -. ps -> A. x -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   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:  ax5e  1841  19.23t  2079  19.23tOLD  2218  xfree2  29304  bj-exalims  32613