Theorem imbibi 252

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.)

Assertion

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

Proof of Theorem imbibi

Step	Hyp	Ref	Expression
1		pm5.4 249	. . 3 ⊢ ((𝜑 → (𝜑 → 𝜓)) ↔ (𝜑 → 𝜓))
2		imbi2 237	. . 3 ⊢ (((𝜑 → 𝜓) ↔ 𝜒) → ((𝜑 → (𝜑 → 𝜓)) ↔ (𝜑 → 𝜒)))
3	1, 2	bitr3id 194	. 2 ⊢ (((𝜑 → 𝜓) ↔ 𝜒) → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
4	3	pm5.74rd 183	1 ⊢ (((𝜑 → 𝜓) ↔ 𝜒) → (𝜑 → (𝜓 ↔ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  snssg  3725