Theorem coeq1

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

	Ref	Expression
Assertion	coeq1	|- ( A = B -> ( A o. C ) = ( B o. C ) )

Proof

Step	Hyp	Ref	Expression
1		coss1	|- ( A C_ B -> ( A o. C ) C_ ( B o. C ) )
2		coss1	|- ( B C_ A -> ( B o. C ) C_ ( A o. C ) )
3	1 2	anim12i	|- ( ( A C_ B /\ B C_ A ) -> ( ( A o. C ) C_ ( B o. C ) /\ ( B o. C ) C_ ( A o. C ) ) )
4		eqss	|- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss	|- ( ( A o. C ) = ( B o. C ) <-> ( ( A o. C ) C_ ( B o. C ) /\ ( B o. C ) C_ ( A o. C ) ) )
6	3 4 5	3imtr4i	|- ( A = B -> ( A o. C ) = ( B o. C ) )