Theorem nbbn 385

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.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)

Assertion

Ref	Expression
nbbn	⊢ ((¬ 𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))

Proof of Theorem nbbn

Step	Hyp	Ref	Expression
1		pm5.18 383	. 2 ⊢ ((¬ 𝜑 ↔ 𝜓) ↔ ¬ (¬ 𝜑 ↔ ¬ 𝜓))
2		notbi 321	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
3	1, 2	xchbinxr 337	1 ⊢ ((¬ 𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 208

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 209

This theorem is referenced by:  biass  387  pclem6  1039  xorass  1535  hadbi  1618  canth  7350  qextltlem  13205  onint1  36806  wl-df3xor2  37960  wl-3xornot1  37971  wl-2xor  37974  notbinot1  38575  notbinot2  38579  onsupmaxb  43813