Metamath Proof Explorer

Theorem sylanr1

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

		Ref	Expression
	Hypotheses	sylanr1.1	$⊢ φ \to χ$
		sylanr1.2	$⊢ (ψ \land (χ \land θ)) \to τ$
	Assertion	sylanr1	$⊢ (ψ \land (φ \land θ)) \to τ$

Proof

Step	Hyp	Ref	Expression
1		sylanr1.1	$⊢ φ \to χ$
2		sylanr1.2	$⊢ (ψ \land (χ \land θ)) \to τ$
3	1	anim1i	$⊢ (φ \land θ) \to (χ \land θ)$
4	3 2	sylan2	$⊢ (ψ \land (φ \land θ)) \to τ$