Theorem sseqtrdi

Description: A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004)

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

Proof

Step	Hyp	Ref	Expression
1		sseqtrdi.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		sseqtrdi.2	⊢ 𝐵 = 𝐶
3	2	sseq2i	⊢ ( 𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶 )
4	1 3	sylib	⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 )