Theorem nbn 689

Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)

Hypothesis

Ref	Expression
nbn.1	|- -. ph

Assertion

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

Proof of Theorem nbn

Step	Hyp	Ref	Expression
1		nbn.1	. . 3 |- -. ph
2		bibif 688	. . 3 |- ( -. ph -> ( ( ps <-> ph ) <-> -. ps ) )
3	1, 2	ax-mp 5	. 2 |- ( ( ps <-> ph ) <-> -. ps )
4	3	bicomi 131	1 |- ( -. ps <-> ( ps <-> ph ) )

Syntax hints:   -. wn 3   <-> wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  nbn3  690  nbfal  1354  n0rf  3421  eq0  3427  disj  3457  dm0rn0  4821  reldm0  4822