Theorem biimpexp

Description: A biconditional in the antecedent is the same as two implications. (Contributed by Scott Fenton, 12-Dec-2010)

	Ref	Expression
Assertion	biimpexp	⊢ ( ( ( 𝜑 ↔ 𝜓 ) → 𝜒 ) ↔ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dfbi2	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) )
2	1	imbi1i	⊢ ( ( ( 𝜑 ↔ 𝜓 ) → 𝜒 ) ↔ ( ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) → 𝜒 ) )
3		impexp	⊢ ( ( ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) → 𝜒 ) ↔ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → 𝜒 ) ) )
4	2 3	bitri	⊢ ( ( ( 𝜑 ↔ 𝜓 ) → 𝜒 ) ↔ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → 𝜒 ) ) )