Theorem nbn 542

Description: The negation of a wff is equivalent to the wff's equivalence to falsehood.

Hypothesis

Ref	Expression
nbn.1	|- -. ph

Assertion

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

Proof of Theorem nbn

Step	Hyp	Ref	Expression
1		pm2.21 71	. . 3 |- (-. ps -> (ps -> ph))
2		nbn.1	. . . . 5 |- -. ph
3	2	a1i 7	. . . 4 |- (-. ps -> -. ph)
4	3	pm2.21d 74	. . 3 |- (-. ps -> (ph -> ps))
5	1, 4	impbid 397	. 2 |- (-. ps -> (ps <-> ph))
6		bi1 130	. . 3 |- ((ps <-> ph) -> (ps -> ph))
7	2, 6	mtoi 94	. 2 |- ((ps <-> ph) -> -. ps)
8	5, 7	impbi 139	1 |- (-. ps <-> (ps <-> ph))

Syntax hints:  -. wn 1   <-> wb 127

This theorem is referenced by:  n0f 1713  disj 1733  dm0rn0 2549  reldm0 2550  intirr 2628

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

This theorem depends on definitions:  df-bi 128  df-an 198

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