Theorem imbibi 394

Description: The antecedent of one side of a biconditional can be moved out of the biconditional to become the antecedent of the remaining biconditional. (Contributed by BJ, 1-Jan-2025.) (Proof shortened by Wolf Lammen, 5-Jan-2025.) (Proof shortened by Garrett Katz, 15-Jun-2026.)

Assertion

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

Proof of Theorem imbibi

Step	Hyp	Ref	Expression
1		bitr3 354	. 2 ⊢ (((𝜑 → 𝜓) ↔ 𝜓) → (((𝜑 → 𝜓) ↔ 𝜒) → (𝜓 ↔ 𝜒)))
2		pm5.5 363	. 2 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ 𝜓))
3	1, 2	syl11 33	1 ⊢ (((𝜑 → 𝜓) ↔ 𝜒) → (𝜑 → (𝜓 ↔ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 208

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 209

This theorem is referenced by:  snssg  4742  regsfromregtco  36895