Theorem iman

Description: Implication in terms of conjunction and negation. Theorem 3.4(27) of Stoll p. 176. (Contributed by NM, 12-Mar-1993) (Proof shortened by Wolf Lammen, 30-Oct-2012)

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

Proof

Step	Hyp	Ref	Expression
1		notnotb	⊢ ( 𝜓 ↔ ¬ ¬ 𝜓 )
2	1	imbi2i	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ¬ ¬ 𝜓 ) )
3		imnan	⊢ ( ( 𝜑 → ¬ ¬ 𝜓 ) ↔ ¬ ( 𝜑 ∧ ¬ 𝜓 ) )
4	2 3	bitri	⊢ ( ( 𝜑 → 𝜓 ) ↔ ¬ ( 𝜑 ∧ ¬ 𝜓 ) )