Theorem brrabga

Description: The law of concretion for operation class abstraction. (Contributed by Peter Mazsa, 24-Oct-2022)

	Ref	Expression
Hypotheses	brrabga.1	|- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )
	brrabga.2	|- R = { <. <. x , y >. , z >. | ph }
Assertion	brrabga	|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <. A , B >. R C <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		brrabga.1	|- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )
2		brrabga.2	|- R = { <. <. x , y >. , z >. | ph }
3		df-br	|- ( <. A , B >. R C <-> <. <. A , B >. , C >. e. R )
4	2	eleq2i	|- ( <. <. A , B >. , C >. e. R <-> <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } )
5	3 4	bitri	|- ( <. A , B >. R C <-> <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } )
6	1	eloprabga	|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> ps ) )
7	5 6	syl5bb	|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <. A , B >. R C <-> ps ) )