Metamath Proof Explorer

Theorem sylan2d

Description: A syllogism deduction. (Contributed by NM, 15-Dec-2004)

		Ref	Expression
	Hypotheses	sylan2d.1	$⊢ φ \to (ψ \to χ)$
		sylan2d.2	$⊢ φ \to ((θ \land χ) \to τ)$
	Assertion	sylan2d	$⊢ φ \to ((θ \land ψ) \to τ)$

Proof

Step	Hyp	Ref	Expression
1		sylan2d.1	$⊢ φ \to (ψ \to χ)$
2		sylan2d.2	$⊢ φ \to ((θ \land χ) \to τ)$
3	2	ancomsd	$⊢ φ \to ((χ \land θ) \to τ)$
4	1 3	syland	$⊢ φ \to ((ψ \land θ) \to τ)$
5	4	ancomsd	$⊢ φ \to ((θ \land ψ) \to τ)$