Theorem coeq2

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

	Ref	Expression
Assertion	coeq2	$$⊢A=B\to C\circ A=C\circ B$$

Proof

Step	Hyp	Ref	Expression
1		coss2	$$⊢A\subseteq B\to C\circ A\subseteq C\circ B$$
2		coss2	$$⊢B\subseteq A\to C\circ B\subseteq C\circ A$$
3	1 2	anim12i	$$⊢\left(A\subseteq B\wedge B\subseteq A\right)\to \left(C\circ A\subseteq C\circ B\wedge C\circ B\subseteq C\circ A\right)$$
4		eqss	$$⊢A=B\leftrightarrow \left(A\subseteq B\wedge B\subseteq A\right)$$
5		eqss	$$⊢C\circ A=C\circ B\leftrightarrow \left(C\circ A\subseteq C\circ B\wedge C\circ B\subseteq C\circ A\right)$$
6	3 4 5	3imtr4i	$$⊢A=B\to C\circ A=C\circ B$$