Theorem impbid21d

Description: Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013)

	Ref	Expression
Hypotheses	impbid21d.1	⊢ ( 𝜓 → ( 𝜒 → 𝜃 ) )
	impbid21d.2	⊢ ( 𝜑 → ( 𝜃 → 𝜒 ) )
Assertion	impbid21d	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ↔ 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		impbid21d.1	⊢ ( 𝜓 → ( 𝜒 → 𝜃 ) )
2		impbid21d.2	⊢ ( 𝜑 → ( 𝜃 → 𝜒 ) )
3		impbi	⊢ ( ( 𝜒 → 𝜃 ) → ( ( 𝜃 → 𝜒 ) → ( 𝜒 ↔ 𝜃 ) ) )
4	1 2 3	syl2imc	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ↔ 𝜃 ) ) )