Theorem coeq1

Description: Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997)

	Ref	Expression
Assertion	coeq1	$$⊢A=B\to A\circ C=B\circ C$$

Proof

Step	Hyp	Ref	Expression
1		coss1	$$⊢A\subseteq B\to A\circ C\subseteq B\circ C$$
2		coss1	$$⊢B\subseteq A\to B\circ C\subseteq A\circ C$$
3	1 2	anim12i	$$⊢\left(A\subseteq B\wedge B\subseteq A\right)\to \left(A\circ C\subseteq B\circ C\wedge B\circ C\subseteq A\circ C\right)$$
4		eqss	$$⊢A=B\leftrightarrow \left(A\subseteq B\wedge B\subseteq A\right)$$
5		eqss	$$⊢A\circ C=B\circ C\leftrightarrow \left(A\circ C\subseteq B\circ C\wedge B\circ C\subseteq A\circ C\right)$$
6	3 4 5	3imtr4i	$$⊢A=B\to A\circ C=B\circ C$$