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	⊢ ((¬ φ ↔ ψ) ↔ ¬ (φ ↔ ψ))

Proof of Theorem nbbn

Step	Hyp	Ref	Expression
1		xor3 346	. 2 ⊢ (¬ (φ ↔ ψ) ↔ (φ ↔ ¬ ψ))
2		con2bi 318	. 2 ⊢ ((φ ↔ ¬ ψ) ↔ (ψ ↔ ¬ φ))
3		bicom 191	. 2 ⊢ ((ψ ↔ ¬ φ) ↔ (¬ φ ↔ ψ))
4	1, 2, 3	3bitrri 263	1 ⊢ ((¬ φ ↔ ψ) ↔ ¬ (φ ↔ ψ))

Syntax hints:  ¬ wn 3   ↔ wb 176

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

This theorem depends on definitions:  df-bi 177

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