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Metamath Proof Explorer

Theorem 3imtr3d

Description: More general version of 3imtr3i . Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996)

Ref Expression
Hypotheses 3imtr3d.1 ⊢ φ → ψ → χ
3imtr3d.2 ⊢ φ → ψ ↔ θ
3imtr3d.3 ⊢ φ → χ ↔ τ
Assertion 3imtr3d ⊢ φ → θ → τ

Proof

Step Hyp Ref Expression
1 3imtr3d.1 ⊢ φ → ψ → χ
2 3imtr3d.2 ⊢ φ → ψ ↔ θ
3 3imtr3d.3 ⊢ φ → χ ↔ τ
4 1 3 sylibd ⊢ φ → ψ → τ
5 2 4 sylbird ⊢ φ → θ → τ