Theorem sstrd

Description: Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004)

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

Proof

Step	Hyp	Ref	Expression
1		sstrd.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		sstrd.2	⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )
3		sstr	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶 ) → 𝐴 ⊆ 𝐶 )
4	1 2 3	syl2anc	⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 )