Theorem 3sstr3g

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000)

	Ref	Expression
Hypotheses	3sstr3g.1	$$⊢\phi \to A\subseteq B$$
	3sstr3g.2	$$⊢A=C$$
	3sstr3g.3	$$⊢B=D$$
Assertion	3sstr3g	$$⊢\phi \to C\subseteq D$$

Proof

Step	Hyp	Ref	Expression
1		3sstr3g.1	$$⊢\phi \to A\subseteq B$$
2		3sstr3g.2	$$⊢A=C$$
3		3sstr3g.3	$$⊢B=D$$
4	2 3	sseq12i	$$⊢A\subseteq B\leftrightarrow C\subseteq D$$
5	1 4	sylib	$$⊢\phi \to C\subseteq D$$