Theorem nbn2 334

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		pm5.501 330	. 2 |- (-. ph -> (-. ps <-> (-. ph <-> -. ps)))
2		notbi 286	. 2 |- ((ph <-> ps) <-> (-. ph <-> -. ps))
3	1, 2	syl6bbr 254	1 |- (-. ph -> (-. ps <-> (ph <-> ps)))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  bibif  335  pm5.21im  338  pm5.18  345  biass  348