Theorem notbi 655

Description: Equivalence property for negation. Closed form. (Contributed by BJ, 27-Jan-2020.)

Assertion

Ref	Expression
notbi	|- ( ( ph <-> ps ) -> ( -. ph <-> -. ps ) )

Proof of Theorem notbi

Step	Hyp	Ref	Expression
1		bi2 129	. . 3 |- ( ( ph <-> ps ) -> ( ps -> ph ) )
2	1	con3d 620	. 2 |- ( ( ph <-> ps ) -> ( -. ph -> -. ps ) )
3		bi1 117	. . 3 |- ( ( ph <-> ps ) -> ( ph -> ps ) )
4	3	con3d 620	. 2 |- ( ( ph <-> ps ) -> ( -. ps -> -. ph ) )
5	2, 4	impbid 128	1 |- ( ( ph <-> ps ) -> ( -. ph <-> -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  notbid  656  notbii  657  ifbi  3487