Metamath Proof Explorer

Theorem 3bior1fd

Description: A disjunction is equivalent to a threefold disjunction with single falsehood, analogous to biorf . (Contributed by Alexander van der Vekens, 8-Sep-2017)

Ref Expression
Hypothesis 3biorfd.1 ⊢ φ → ¬ θ
Assertion 3bior1fd ⊢ φ → χ ∨ ψ ↔ θ ∨ χ ∨ ψ

Proof

Step Hyp Ref Expression
1 3biorfd.1 ⊢ φ → ¬ θ
2 biorf ⊢ ¬ θ → χ ∨ ψ ↔ θ ∨ χ ∨ ψ
3 1 2 syl ⊢ φ → χ ∨ ψ ↔ θ ∨ χ ∨ ψ
4 3orass ⊢ θ ∨ χ ∨ ψ ↔ θ ∨ χ ∨ ψ
5 3 4 bitr4di ⊢ φ → χ ∨ ψ ↔ θ ∨ χ ∨ ψ