Theorem niabn 917

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 447	. 2 ⊢ ((χ ∧ ψ) → ψ)
2		niabn.1	. . 3 ⊢ φ
3	2	pm2.24i 136	. 2 ⊢ (¬ φ → ψ)
4	1, 3	pm5.21ni 341	1 ⊢ (¬ ψ → ((χ ∧ ψ) ↔ ¬ φ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358

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 177  df-an 360

This theorem is referenced by:  ninba  927