Theorem sseq2

Description: Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998)

	Ref	Expression
Assertion	sseq2	$$⊢A=B\to \left(C\subseteq A\leftrightarrow C\subseteq B\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqss	$$⊢A=B\leftrightarrow \left(A\subseteq B\wedge B\subseteq A\right)$$
2		sstr2	$$⊢C\subseteq A\to \left(A\subseteq B\to C\subseteq B\right)$$
3	2	com12	$$⊢A\subseteq B\to \left(C\subseteq A\to C\subseteq B\right)$$
4		sstr2	$$⊢C\subseteq B\to \left(B\subseteq A\to C\subseteq A\right)$$
5	4	com12	$$⊢B\subseteq A\to \left(C\subseteq B\to C\subseteq A\right)$$
6	3 5	anbiim	$$⊢\left(A\subseteq B\wedge B\subseteq A\right)\to \left(C\subseteq A\leftrightarrow C\subseteq B\right)$$
7	1 6	sylbi	$$⊢A=B\to \left(C\subseteq A\leftrightarrow C\subseteq B\right)$$