Theorem 3sstr4d

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995) (Proof shortened by Eric Schmidt, 26-Jan-2007)

	Ref	Expression
Hypotheses	3sstr4d.1	$$⊢\phi \to A\subseteq B$$
	3sstr4d.2	$$⊢\phi \to C=A$$
	3sstr4d.3	$$⊢\phi \to D=B$$
Assertion	3sstr4d	$$⊢\phi \to C\subseteq D$$

Proof

Step	Hyp	Ref	Expression
1		3sstr4d.1	$$⊢\phi \to A\subseteq B$$
2		3sstr4d.2	$$⊢\phi \to C=A$$
3		3sstr4d.3	$$⊢\phi \to D=B$$
4	2 3	sseq12d	$$⊢\phi \to \left(C\subseteq D\leftrightarrow A\subseteq B\right)$$
5	1 4	mpbird	$$⊢\phi \to C\subseteq D$$