Theorem biimpexp 33659

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

Assertion

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

Proof of Theorem biimpexp

Step	Hyp	Ref	Expression
1		dfbi2 475	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
2	1	imbi1i 350	. 2 ⊢ (((𝜑 ↔ 𝜓) → 𝜒) ↔ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → 𝜒))
3		impexp 451	. 2 ⊢ ((((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → 𝜒) ↔ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → 𝜒)))
4	2, 3	bitri 274	1 ⊢ (((𝜑 ↔ 𝜓) → 𝜒) ↔ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → 𝜒)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 397

This theorem is referenced by:  axextdfeq  33773