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Metamath Proof Explorer

Theorem syl2an

Description: A double syllogism inference. For an implication-only version, see syl2im . (Contributed by NM, 31-Jan-1997)

Ref Expression
Hypotheses syl2an.1 ⊢ φ → ψ
syl2an.2 ⊢ τ → χ
syl2an.3 ⊢ ψ ∧ χ → θ
Assertion syl2an ⊢ φ ∧ τ → θ

Proof

Step Hyp Ref Expression
1 syl2an.1 ⊢ φ → ψ
2 syl2an.2 ⊢ τ → χ
3 syl2an.3 ⊢ ψ ∧ χ → θ
4 1 3 sylan ⊢ φ ∧ χ → θ
5 2 4 sylan2 ⊢ φ ∧ τ → θ