Theorem nbn2 645

Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.)

Assertion

Ref	Expression
nbn2	|- ( -. ph -> ( -. ps <-> ( ph <-> ps ) ) )

Proof of Theorem nbn2

Step	Hyp	Ref	Expression
1		pm5.21im 644	. 2 |- ( -. ph -> ( -. ps -> ( ph <-> ps ) ) )
2		bi2 128	. . 3 |- ( ( ph <-> ps ) -> ( ps -> ph ) )
3		mtt 642	. . 3 |- ( -. ph -> ( -. ps <-> ( ps -> ph ) ) )
4	2, 3	syl5ibr 154	. 2 |- ( -. ph -> ( ( ph <-> ps ) -> -. ps ) )
5	1, 4	impbid 127	1 |- ( -. ph -> ( -. ps <-> ( ph <-> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  bibif  646  pm5.18dc  810  biassdc  1326