Theorem imnot 355

Description: If a proposition is false, then implying it is equivalent to being false. One of four theorems that can be used to simplify an implication ( ph -> ps ), the other ones being ax-1 6 (true consequent), pm2.21 120 (false antecedent), pm5.5 351 (true antecedent). (Contributed by Mario Carneiro, 26-Apr-2019.) (Proof shortened by Wolf Lammen, 26-May-2019.)

Assertion

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

Proof of Theorem imnot

Step	Hyp	Ref	Expression
1		mtt 354	. 2 |- ( -. ps -> ( -. ph <-> ( ph -> ps ) ) )
2	1	bicomd 213	1 |- ( -. ps -> ( ( ph -> ps ) <-> -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196

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

This theorem is referenced by:  sup0riota  8371  ntrneikb  38392