Theorem bj-nnsn 12945

Description: As far as implying a negated formula is concerned, a formula is equivalent to its double negation. (Contributed by BJ, 24-Nov-2023.)

Assertion

Ref	Expression
bj-nnsn	|- ( ( ph -> -. ps ) <-> ( -. -. ph -> -. ps ) )

Proof of Theorem bj-nnsn

Step	Hyp	Ref	Expression
1		con3 631	. . . 4 |- ( ( ph -> -. ps ) -> ( -. -. ps -> -. ph ) )
2	1	con3d 620	. . 3 |- ( ( ph -> -. ps ) -> ( -. -. ph -> -. -. -. ps ) )
3		notnotnot 623	. . 3 |- ( -. -. -. ps <-> -. ps )
4	2, 3	syl6ib 160	. 2 |- ( ( ph -> -. ps ) -> ( -. -. ph -> -. ps ) )
5		notnot 618	. . 3 |- ( ph -> -. -. ph )
6	5	imim1i 60	. 2 |- ( ( -. -. ph -> -. ps ) -> ( ph -> -. ps ) )
7	4, 6	impbii 125	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: (None)