Theorem 3impexp

Description: Version of impexp for a triple conjunction. (Contributed by Alan Sare, 31-Dec-2011)

	Ref	Expression
Assertion	3impexp	⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) )
2	1	3expd	⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) )
3		id	⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) )
4	3	3impd	⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) )
5	2 4	impbii	⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) )