Theorem nbn2 360

Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)

Assertion

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

Proof of Theorem nbn2

Step	Hyp	Ref	Expression
1		pm5.501 356	. 2 |- ( -. ph -> ( -. ps <-> ( -. ph <-> -. ps ) ) )
2		notbi 309	. 2 |- ( ( ph <-> ps ) <-> ( -. ph <-> -. ps ) )
3	1, 2	syl6bbr 278	1 |- ( -. ph -> ( -. ps <-> ( ph <-> ps ) ) )

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:  bibif  361  pm5.21im  364  pm5.18  371  biass  374  sadadd2lem2  15172  isclo  20891