Theorem nbn2 720

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

Assertion

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

Proof of Theorem nbn2

Step	Hyp	Ref	Expression
1		pm5.21 676	. . 3 |- ((-. ph /\ -. ps) -> (ph <-> ps))
2	1	ex 373	. 2 |- (-. ph -> (-. ps -> (ph <-> ps)))
3		pm4.11 521	. . . 4 |- ((ph <-> ps) <-> (-. ph <-> -. ps))
4	3	biimp 151	. . 3 |- ((ph <-> ps) -> (-. ph <-> -. ps))
5	4	biimpcd 155	. 2 |- (-. ph -> ((ph <-> ps) -> -. ps))
6	2, 5	impbid 515	1 |- (-. ph -> (-. ps <-> (ph <-> ps)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146

This theorem is referenced by:  nbn 721  biass 743

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225

Copyright terms: Public domain