Theorem syl2and

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

	Ref	Expression
Hypotheses	syl2and.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	syl2and.2	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
	syl2and.3	⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜏 ) → 𝜂 ) )
Assertion	syl2and	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜃 ) → 𝜂 ) )

Proof

Step	Hyp	Ref	Expression
1		syl2and.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		syl2and.2	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
3		syl2and.3	⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜏 ) → 𝜂 ) )
4	2 3	sylan2d	⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜃 ) → 𝜂 ) )
5	1 4	syland	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜃 ) → 𝜂 ) )