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)

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

Proof

Step Hyp Ref Expression
1 pm5.4 ⊢ φ → φ → ψ ↔ φ → ψ
2 imbi2 ⊢ φ → ψ ↔ χ → φ → φ → ψ ↔ φ → χ
3 1 2 bitr3id ⊢ φ → ψ ↔ χ → φ → ψ ↔ φ → χ
4 3 pm5.74rd ⊢ φ → ψ ↔ χ → φ → ψ ↔ χ