Theorem sylan9ssr

Description: A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004)

	Ref	Expression
Hypotheses	sylan9ssr.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
	sylan9ssr.2	⊢ ( 𝜓 → 𝐵 ⊆ 𝐶 )
Assertion	sylan9ssr	⊢ ( ( 𝜓 ∧ 𝜑 ) → 𝐴 ⊆ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		sylan9ssr.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		sylan9ssr.2	⊢ ( 𝜓 → 𝐵 ⊆ 𝐶 )
3	1 2	sylan9ss	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 ⊆ 𝐶 )
4	3	ancoms	⊢ ( ( 𝜓 ∧ 𝜑 ) → 𝐴 ⊆ 𝐶 )