Theorem nbn2 698

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 697	. 2 |- ( -. ph -> ( -. ps -> ( ph <-> ps ) ) )
2		biimpr 130	. . 3 |- ( ( ph <-> ps ) -> ( ps -> ph ) )
3		mtt 686	. . 3 |- ( -. ph -> ( -. ps <-> ( ps -> ph ) ) )
4	2, 3	imbitrrid 156	. 2 |- ( -. ph -> ( ( ph <-> ps ) -> -. ps ) )
5	1, 4	impbid 129	1 |- ( -. ph -> ( -. ps <-> ( ph <-> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bibif  699  pm5.18dc  884  biassdc  1406