Theorem eloprabg

Description: The law of concretion for operation class abstraction. Compare elopab . (Contributed by NM, 14-Sep-1999) (Revised by David Abernethy, 19-Jun-2012)

	Ref	Expression
Hypotheses	eloprabg.1	|- ( x = A -> ( ph <-> ps ) )
	eloprabg.2	|- ( y = B -> ( ps <-> ch ) )
	eloprabg.3	|- ( z = C -> ( ch <-> th ) )
Assertion	eloprabg	|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> th ) )

Proof

Step	Hyp	Ref	Expression
1		eloprabg.1	|- ( x = A -> ( ph <-> ps ) )
2		eloprabg.2	|- ( y = B -> ( ps <-> ch ) )
3		eloprabg.3	|- ( z = C -> ( ch <-> th ) )
4	1 2 3	syl3an9b	|- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> th ) )
5	4	eloprabga	|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> th ) )