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Metamath Proof Explorer

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 1	eqsstri	⊢ 𝐶 ⊆ 𝐵
5	4 3	sseqtrri	⊢ 𝐶 ⊆ 𝐷