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Metamath Proof Explorer

Theorem ifpbi23d

Description: Equivalence deduction for conditional operator for propositions. Convenience theorem for a frequent case. (Contributed by Wolf Lammen, 28-Apr-2024)

Ref Expression
Hypotheses ifpbi23d.1 ⊢ φ → χ ↔ η
ifpbi23d.2 ⊢ φ → θ ↔ ζ
Assertion ifpbi23d ⊢ φ → if- ψ χ θ ↔ if- ψ η ζ

Proof

Step Hyp Ref Expression
1 ifpbi23d.1 ⊢ φ → χ ↔ η
2 ifpbi23d.2 ⊢ φ → θ ↔ ζ
3 biidd ⊢ φ → ψ ↔ ψ
4 3 1 2 ifpbi123d ⊢ φ → if- ψ χ θ ↔ if- ψ η ζ