Theorem eqsstrd

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004)

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

Proof

Step	Hyp	Ref	Expression
1		eqsstrd.1	$$⊢\phi \to A=B$$
2		eqsstrd.2	$$⊢\phi \to B\subseteq C$$
3	1	sseq1d	$$⊢\phi \to \left(A\subseteq C\leftrightarrow B\subseteq C\right)$$
4	2 3	mpbird	$$⊢\phi \to A\subseteq C$$