Theorem ninba 879

Description: Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)

Hypothesis

Ref	Expression
ninba.1	⊢ 𝜑

Assertion

Ref	Expression
ninba	⊢ (¬ 𝜓 → (¬ 𝜑 ↔ (𝜒 ∧ 𝜓)))

Proof of Theorem ninba

Step	Hyp	Ref	Expression
1		ninba.1	. . 3 ⊢ 𝜑
2	1	niabn 874	. 2 ⊢ (¬ 𝜓 → ((𝜒 ∧ 𝜓) ↔ ¬ 𝜑))
3	2	bicomd 129	1 ⊢ (¬ 𝜓 → (¬ 𝜑 ↔ (𝜒 ∧ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)