Theorem con2bi 343

Description: Contraposition. Theorem *4.12 of [WhiteheadRussell] p. 117. (Contributed by NM, 15-Apr-1995.) (Proof shortened by Wolf Lammen, 3-Jan-2013.)

Assertion

Ref	Expression
con2bi	|- ( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) )

Proof of Theorem con2bi

Step	Hyp	Ref	Expression
1		notbi 309	. 2 |- ( ( ph <-> -. ps ) <-> ( -. ph <-> -. -. ps ) )
2		notnotb 304	. . 3 |- ( ps <-> -. -. ps )
3	2	bibi2i 327	. 2 |- ( ( -. ph <-> ps ) <-> ( -. ph <-> -. -. ps ) )
4		bicom 212	. 2 |- ( ( -. ph <-> ps ) <-> ( ps <-> -. ph ) )
5	1, 3, 4	3bitr2i 288	1 |- ( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) )

Syntax hints:   -. wn 3   <-> 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:  con2bid  344  nbbn  373