Metamath Proof Explorer

Theorem sylanr2

Description: A syllogism inference. (Contributed by NM, 9-Apr-2005)

		Ref	Expression
	Hypotheses	sylanr2.1	$⊢ φ \to θ$
		sylanr2.2	$⊢ (ψ \land (χ \land θ)) \to τ$
	Assertion	sylanr2	$⊢ (ψ \land (χ \land φ)) \to τ$

Proof

Step	Hyp	Ref	Expression
1		sylanr2.1	$⊢ φ \to θ$
2		sylanr2.2	$⊢ (ψ \land (χ \land θ)) \to τ$
3	1	anim2i	$⊢ (χ \land φ) \to (χ \land θ)$
4	3 2	sylan2	$⊢ (ψ \land (χ \land φ)) \to τ$