Theorem nbbn 386

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		pm5.18 384	. 2 ⊢ ((¬ 𝜑 ↔ 𝜓) ↔ ¬ (¬ 𝜑 ↔ ¬ 𝜓))
2		notbi 322	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
3	1, 2	xchbinxr 338	1 ⊢ ((¬ 𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209

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

This proof depends on definitions:  df-bi 210

This theorem is used by:  biass  388  pclem6  1043  xorass  1544  hadbi  1627  nrmod  3839  canth  7376  qextltlem  13337  onint1  37331  wl-df3xor2  38499  wl-3xornot1  38510  wl-2xor  38513  notbinot1  39120  notbinot2  39124  onsupmaxb  44348