Theorem nan

Description: Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003)

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

Proof

Step	Hyp	Ref	Expression
1		impexp	⊢ ( ( ( 𝜑 ∧ 𝜓 ) → ¬ 𝜒 ) ↔ ( 𝜑 → ( 𝜓 → ¬ 𝜒 ) ) )
2		imnan	⊢ ( ( 𝜓 → ¬ 𝜒 ) ↔ ¬ ( 𝜓 ∧ 𝜒 ) )
3	2	imbi2i	⊢ ( ( 𝜑 → ( 𝜓 → ¬ 𝜒 ) ) ↔ ( 𝜑 → ¬ ( 𝜓 ∧ 𝜒 ) ) )
4	1 3	bitr2i	⊢ ( ( 𝜑 → ¬ ( 𝜓 ∧ 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) → ¬ 𝜒 ) )