Theorem 3sstr4g

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994) (Proof shortened by Eric Schmidt, 26-Jan-2007)

	Ref	Expression
Hypotheses	3sstr4g.1	$$⊢\phi \to A\subseteq B$$
	3sstr4g.2	$$⊢C=A$$
	3sstr4g.3	$$⊢D=B$$
Assertion	3sstr4g	$$⊢\phi \to C\subseteq D$$

Proof

Step	Hyp	Ref	Expression
1		3sstr4g.1	$$⊢\phi \to A\subseteq B$$
2		3sstr4g.2	$$⊢C=A$$
3		3sstr4g.3	$$⊢D=B$$
4	2 3	sseq12i	$$⊢C\subseteq D\leftrightarrow A\subseteq B$$
5	1 4	sylibr	$$⊢\phi \to C\subseteq D$$