Theorem impbid2

Description: Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007) (Proof shortened by Wolf Lammen, 27-Sep-2013)

	Ref	Expression
Hypotheses	impbid2.1	⊢ ( 𝜓 → 𝜒 )
	impbid2.2	⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) )
Assertion	impbid2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		impbid2.1	⊢ ( 𝜓 → 𝜒 )
2		impbid2.2	⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) )
3	2 1	impbid1	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜓 ) )
4	3	bicomd	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )