Theorem coeq2

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

	Ref	Expression
Assertion	coeq2	|- ( A = B -> ( C o. A ) = ( C o. B ) )

Proof

Step	Hyp	Ref	Expression
1		coss2	|- ( A C_ B -> ( C o. A ) C_ ( C o. B ) )
2		coss2	|- ( B C_ A -> ( C o. B ) C_ ( C o. A ) )
3	1 2	anim12i	|- ( ( A C_ B /\ B C_ A ) -> ( ( C o. A ) C_ ( C o. B ) /\ ( C o. B ) C_ ( C o. A ) ) )
4		eqss	|- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss	|- ( ( C o. A ) = ( C o. B ) <-> ( ( C o. A ) C_ ( C o. B ) /\ ( C o. B ) C_ ( C o. A ) ) )
6	3 4 5	3imtr4i	|- ( A = B -> ( C o. A ) = ( C o. B ) )