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Metamath Proof Explorer

Theorem syl3anc

Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012)

Ref Expression
Hypotheses syl3anc.1 ⊢ φ → ψ
syl3anc.2 ⊢ φ → χ
syl3anc.3 ⊢ φ → θ
syl3anc.4 ⊢ ψ ∧ χ ∧ θ → τ
Assertion syl3anc ⊢ φ → τ

Proof

Step Hyp Ref Expression
1 syl3anc.1 ⊢ φ → ψ
2 syl3anc.2 ⊢ φ → χ
3 syl3anc.3 ⊢ φ → θ
4 syl3anc.4 ⊢ ψ ∧ χ ∧ θ → τ
5 1 2 3 3jca ⊢ φ → ψ ∧ χ ∧ θ
6 5 4 syl ⊢ φ → τ