Theorem niabn 969

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

Hypothesis

Ref	Expression
niabn.1	⊢ 𝜑

Assertion

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

Proof of Theorem niabn

Step	Hyp	Ref	Expression
1		simpr 110	. 2 ⊢ ((𝜒 ∧ 𝜓) → 𝜓)
2		niabn.1	. . 3 ⊢ 𝜑
3	2	pm2.24i 624	. 2 ⊢ (¬ 𝜑 → 𝜓)
4	1, 3	pm5.21ni 704	1 ⊢ (¬ 𝜓 → ((𝜒 ∧ 𝜓) ↔ ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  ninba  974