Theorem impbii

Description: Infer an equivalence from an implication and its converse. Inference associated with impbi . (Contributed by NM, 29-Dec-1992)

	Ref	Expression
Hypotheses	impbii.1	⊢ ( 𝜑 → 𝜓 )
	impbii.2	⊢ ( 𝜓 → 𝜑 )
Assertion	impbii	⊢ ( 𝜑 ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		impbii.1	⊢ ( 𝜑 → 𝜓 )
2		impbii.2	⊢ ( 𝜓 → 𝜑 )
3		impbi	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ( 𝜑 ↔ 𝜓 ) ) )
4	1 2 3	mp2	⊢ ( 𝜑 ↔ 𝜓 )