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Metamath Proof Explorer

Theorem 3bitr3rd

Description: Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006)

Ref Expression
Hypotheses 3bitr3d.1 ⊢ φ → ψ ↔ χ
3bitr3d.2 ⊢ φ → ψ ↔ θ
3bitr3d.3 ⊢ φ → χ ↔ τ
Assertion 3bitr3rd ⊢ φ → τ ↔ θ

Proof

Step Hyp Ref Expression
1 3bitr3d.1 ⊢ φ → ψ ↔ χ
2 3bitr3d.2 ⊢ φ → ψ ↔ θ
3 3bitr3d.3 ⊢ φ → χ ↔ τ
4 1 2 bitr3d ⊢ φ → χ ↔ θ
5 3 4 bitr3d ⊢ φ → τ ↔ θ