Theorem 3sstr3d

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

	Ref	Expression
Hypotheses	3sstr3d.1	$$⊢\phi \to A\subseteq B$$
	3sstr3d.2	$$⊢\phi \to A=C$$
	3sstr3d.3	$$⊢\phi \to B=D$$
Assertion	3sstr3d	$$⊢\phi \to C\subseteq D$$

Proof

Step	Hyp	Ref	Expression
1		3sstr3d.1	$$⊢\phi \to A\subseteq B$$
2		3sstr3d.2	$$⊢\phi \to A=C$$
3		3sstr3d.3	$$⊢\phi \to B=D$$
4	2 3	sseq12d	$$⊢\phi \to \left(A\subseteq B\leftrightarrow C\subseteq D\right)$$
5	1 4	mpbid	$$⊢\phi \to C\subseteq D$$