Theorem nbn2 351

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		pm5.501 347	. 2 |- ( -. ph -> ( -. ps <-> ( -. ph <-> -. ps ) ) )
2		notbi 301	. 2 |- ( ( ph <-> ps ) <-> ( -. ph <-> -. ps ) )
3	1, 2	syl6bbr 271	1 |- ( -. ph -> ( -. ps <-> ( ph <-> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189

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 190

This theorem is referenced by:  bibif  352  pm5.21im  355  pm5.18  362  biass  365  sadadd2lem2  14473  isclo  20152