Theorem sseqtrd

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

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

Proof

Step	Hyp	Ref	Expression
1		sseqtrd.1	$$⊢\phi \to A\subseteq B$$
2		sseqtrd.2	$$⊢\phi \to B=C$$
3	2	sseq2d	$$⊢\phi \to \left(A\subseteq B\leftrightarrow A\subseteq C\right)$$
4	1 3	mpbid	$$⊢\phi \to A\subseteq C$$