Theorem alimex 1700

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 eximal 1663. (Contributed by BJ, 12-May-2019.)

Assertion

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

Proof of Theorem alimex

Step	Hyp	Ref	Expression
1		alex 1695	. . 3 |- ( A. x ps <-> -. E. x -. ps )
2	1	imbi2i 314	. 2 |- ( ( ph -> A. x ps ) <-> ( ph -> -. E. x -. ps ) )
3		con2b 336	. 2 |- ( ( ph -> -. E. x -. ps ) <-> ( E. x -. ps -> -. ph ) )
4	2, 3	bitri 253	1 |- ( ( ph -> A. x ps ) <-> ( E. x -. ps -> -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   A.wal 1436   E.wex 1660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679

This theorem depends on definitions:  df-bi 189  df-ex 1661

This theorem is referenced by: (None)