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

Metamath Proof Explorer

Theorem syl2and

Description: A syllogism deduction. (Contributed by NM, 15-Dec-2004)

Ref Expression
Hypotheses syl2and.1 ⊢ φ → ψ → χ
syl2and.2 ⊢ φ → θ → τ
syl2and.3 ⊢ φ → χ ∧ τ → η
Assertion syl2and ⊢ φ → ψ ∧ θ → η

Proof

Step Hyp Ref Expression
1 syl2and.1 ⊢ φ → ψ → χ
2 syl2and.2 ⊢ φ → θ → τ
3 syl2and.3 ⊢ φ → χ ∧ τ → η
4 2 3 sylan2d ⊢ φ → χ ∧ θ → η
5 1 4 syland ⊢ φ → ψ ∧ θ → η