Theorem nbbn 347

Description: Move negation outside of biconditional. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by NM, 27-Jun-2002.) (Proof shortened by Wolf Lammen, 20-Sep-2013.)

Assertion

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

Proof of Theorem nbbn

Step	Hyp	Ref	Expression
1		xor3 346	. 2 |- (-. (ph <-> ps) <-> (ph <-> -. ps))
2		con2bi 318	. 2 |- ((ph <-> -. ps) <-> (ps <-> -. ph))
3		bicom 191	. 2 |- ((ps <-> -. ph) <-> (-. ph <-> ps))
4	1, 2, 3	3bitrri 263	1 |- ((-. ph <-> ps) <-> -. (ph <-> ps))

This proof depends on syntax axioms:  -. wn 3   <-> wb 176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177

This theorem is used by:  biass  348  pclem6  896  xorass  1308  xorneg1  1311  trubifal  1351  hadbi  1387  mpto2OLD  1535