Metamath Proof Explorer

Theorem syld3an2

Description: A syllogism inference. (Contributed by NM, 20-May-2007)

		Ref	Expression
	Hypotheses	syld3an2.1	$⊢ (φ \land χ \land θ) \to ψ$
		syld3an2.2	$⊢ (φ \land ψ \land θ) \to τ$
	Assertion	syld3an2	$⊢ (φ \land χ \land θ) \to τ$

Proof

Step	Hyp	Ref	Expression
1		syld3an2.1	$⊢ (φ \land χ \land θ) \to ψ$
2		syld3an2.2	$⊢ (φ \land ψ \land θ) \to τ$
3		simp1	$⊢ (φ \land χ \land θ) \to φ$
4		simp3	$⊢ (φ \land χ \land θ) \to θ$
5	3 1 4 2	syl3anc	$⊢ (φ \land χ \land θ) \to τ$