Theorem impbida

Description: Deduce an equivalence from two implications. Variant of impbid . (Contributed by NM, 17-Feb-2007)

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

Proof

Step	Hyp	Ref	Expression
1		impbida.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
2		impbida.2	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜓 )
3	1	ex	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
4	2	ex	⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) )
5	3 4	impbid	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )