Theorem eqsstrid

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

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

Proof

Step	Hyp	Ref	Expression
1		eqsstrid.1	⊢ 𝐴 = 𝐵
2		eqsstrid.2	⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )
3	1	sseq1i	⊢ ( 𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶 )
4	2 3	sylibr	⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 )