This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem 3imtr3i

Description: A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994)

Ref Expression
Hypotheses 3imtr3.1 ⊢ φ → ψ
3imtr3.2 ⊢ φ ↔ χ
3imtr3.3 ⊢ ψ ↔ θ
Assertion 3imtr3i ⊢ χ → θ

Proof

Step Hyp Ref Expression
1 3imtr3.1 ⊢ φ → ψ
2 3imtr3.2 ⊢ φ ↔ χ
3 3imtr3.3 ⊢ ψ ↔ θ
4 2 1 sylbir ⊢ χ → ψ
5 4 3 sylib ⊢ χ → θ