Theorem eqsstrrd

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004)

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

Proof

Step	Hyp	Ref	Expression
1		eqsstrrd.1	⊢ ( 𝜑 → 𝐵 = 𝐴 )
2		eqsstrrd.2	⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )
3	1	eqcomd	⊢ ( 𝜑 → 𝐴 = 𝐵 )
4	3 2	eqsstrd	⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 )