Theorem sylan2d

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

	Ref	Expression
Hypotheses	sylan2d.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	sylan2d.2	⊢ ( 𝜑 → ( ( 𝜃 ∧ 𝜒 ) → 𝜏 ) )
Assertion	sylan2d	⊢ ( 𝜑 → ( ( 𝜃 ∧ 𝜓 ) → 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		sylan2d.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		sylan2d.2	⊢ ( 𝜑 → ( ( 𝜃 ∧ 𝜒 ) → 𝜏 ) )
3	2	ancomsd	⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜃 ) → 𝜏 ) )
4	1 3	syland	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜃 ) → 𝜏 ) )
5	4	ancomsd	⊢ ( 𝜑 → ( ( 𝜃 ∧ 𝜓 ) → 𝜏 ) )