Theorem ninba 913

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 908	. 2 ⊢ (¬ 𝜓 → ((𝜒 ∧ 𝜓) ↔ ¬ 𝜑))
3	2	bicomd 139	1 ⊢ (¬ 𝜓 → (¬ 𝜑 ↔ (𝜒 ∧ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)