Theorem 3sstr4i

Description: Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996) (Proof shortened by Eric Schmidt, 26-Jan-2007)

	Ref	Expression
Hypotheses	3sstr4.1	⊢ 𝐴 ⊆ 𝐵
	3sstr4.2	⊢ 𝐶 = 𝐴
	3sstr4.3	⊢ 𝐷 = 𝐵
Assertion	3sstr4i	⊢ 𝐶 ⊆ 𝐷

Proof

Step	Hyp	Ref	Expression
1		3sstr4.1	⊢ 𝐴 ⊆ 𝐵
2		3sstr4.2	⊢ 𝐶 = 𝐴
3		3sstr4.3	⊢ 𝐷 = 𝐵
4	2 3	sseq12i	⊢ ( 𝐶 ⊆ 𝐷 ↔ 𝐴 ⊆ 𝐵 )
5	1 4	mpbir	⊢ 𝐶 ⊆ 𝐷