Theorem nbbn 383

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 382	. 2 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
2		con2bi 353	. 2 ⊢ ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))
3		bicom 222	. 2 ⊢ ((𝜓 ↔ ¬ 𝜑) ↔ (¬ 𝜑 ↔ 𝜓))
4	1, 2, 3	3bitrri 298	1 ⊢ ((¬ 𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 206

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 207

This theorem is referenced by:  biass  384  pclem6  1028  xorass  1515  hadbi  1598  canth  7385  qextltlem  13244  onint1  36450  wl-df3xor2  37470  wl-3xornot1  37481  wl-2xor  37484  notbinot1  38086  notbinot2  38090  onsupmaxb  43251