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		sstr2	$$⊢C\subseteq A\to \left(A\subseteq B\to C\subseteq B\right)$$
2	1	com12	$$⊢A\subseteq B\to \left(C\subseteq A\to C\subseteq B\right)$$
3		sstr2	$$⊢C\subseteq B\to \left(B\subseteq A\to C\subseteq A\right)$$
4	3	com12	$$⊢B\subseteq A\to \left(C\subseteq B\to C\subseteq A\right)$$
5	2 4	anim12i	$$⊢\left(A\subseteq B\wedge B\subseteq A\right)\to \left(\left(C\subseteq A\to C\subseteq B\right)\wedge \left(C\subseteq B\to C\subseteq A\right)\right)$$
6		eqss	$$⊢A=B\leftrightarrow \left(A\subseteq B\wedge B\subseteq A\right)$$
7		dfbi2	$$⊢\left(C\subseteq A\leftrightarrow C\subseteq B\right)\leftrightarrow \left(\left(C\subseteq A\to C\subseteq B\right)\wedge \left(C\subseteq B\to C\subseteq A\right)\right)$$
8	5 6 7	3imtr4i	$$⊢A=B\to \left(C\subseteq A\leftrightarrow C\subseteq B\right)$$