Metamath Proof Explorer

Theorem imbibi

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)

Ref Expression
Assertion imbibi ⊢ φ → ψ ↔ χ → φ → ψ ↔ χ

Proof

Step Hyp Ref Expression
1 bitr3 ⊢ φ → ψ ↔ ψ → φ → ψ ↔ χ → ψ ↔ χ
2 pm5.5 ⊢ φ → φ → ψ ↔ ψ
3 1 2 syl11 ⊢ φ → ψ ↔ χ → φ → ψ ↔ χ