Theorem 3impexpbicom

Description: Version of 3impexp where in addition the consequent is commuted. (Contributed by Alan Sare, 31-Dec-2011)

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

Proof

Step	Hyp	Ref	Expression
1		bicom	⊢ ( ( 𝜃 ↔ 𝜏 ) ↔ ( 𝜏 ↔ 𝜃 ) )
2		imbi2	⊢ ( ( ( 𝜃 ↔ 𝜏 ) ↔ ( 𝜏 ↔ 𝜃 ) ) → ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜏 ↔ 𝜃 ) ) ) )
3	2	biimpcd	⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) → ( ( ( 𝜃 ↔ 𝜏 ) ↔ ( 𝜏 ↔ 𝜃 ) ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜏 ↔ 𝜃 ) ) ) )
4	1 3	mpi	⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜏 ↔ 𝜃 ) ) )
5	4	3expd	⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) → ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) )
6		3impexp	⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜏 ↔ 𝜃 ) ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) )
7	6	biimpri	⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜏 ↔ 𝜃 ) ) )
8	7 1	syl6ibr	⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) )
9	5 8	impbii	⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) )