Theorem impbi

Description: Property of the biconditional connective. (Contributed by NM, 11-May-1999)

	Ref	Expression
Assertion	impbi	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ( 𝜑 ↔ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-bi	⊢ ¬ ( ( ( 𝜑 ↔ 𝜓 ) → ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) ) → ¬ ( ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) → ( 𝜑 ↔ 𝜓 ) ) )
2		simprim	⊢ ( ¬ ( ( ( 𝜑 ↔ 𝜓 ) → ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) ) → ¬ ( ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) → ( 𝜑 ↔ 𝜓 ) ) ) → ( ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) → ( 𝜑 ↔ 𝜓 ) ) )
3	1 2	ax-mp	⊢ ( ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) → ( 𝜑 ↔ 𝜓 ) )
4	3	expi	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ( 𝜑 ↔ 𝜓 ) ) )