Theorem caovord

Description: Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996)

	Ref	Expression
Hypotheses	caovord.1	|- A e. _V
	caovord.2	|- B e. _V
	caovord.3	|- ( z e. S -> ( x R y <-> ( z F x ) R ( z F y ) ) )
Assertion	caovord	|- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) )

Proof

Step	Hyp	Ref	Expression
1		caovord.1	|- A e. _V
2		caovord.2	|- B e. _V
3		caovord.3	|- ( z e. S -> ( x R y <-> ( z F x ) R ( z F y ) ) )
4		oveq1	|- ( z = C -> ( z F A ) = ( C F A ) )
5		oveq1	|- ( z = C -> ( z F B ) = ( C F B ) )
6	4 5	breq12d	|- ( z = C -> ( ( z F A ) R ( z F B ) <-> ( C F A ) R ( C F B ) ) )
7	6	bibi2d	|- ( z = C -> ( ( A R B <-> ( z F A ) R ( z F B ) ) <-> ( A R B <-> ( C F A ) R ( C F B ) ) ) )
8		breq1	|- ( x = A -> ( x R y <-> A R y ) )
9		oveq2	|- ( x = A -> ( z F x ) = ( z F A ) )
10	9	breq1d	|- ( x = A -> ( ( z F x ) R ( z F y ) <-> ( z F A ) R ( z F y ) ) )
11	8 10	bibi12d	|- ( x = A -> ( ( x R y <-> ( z F x ) R ( z F y ) ) <-> ( A R y <-> ( z F A ) R ( z F y ) ) ) )
12		breq2	|- ( y = B -> ( A R y <-> A R B ) )
13		oveq2	|- ( y = B -> ( z F y ) = ( z F B ) )
14	13	breq2d	|- ( y = B -> ( ( z F A ) R ( z F y ) <-> ( z F A ) R ( z F B ) ) )
15	12 14	bibi12d	|- ( y = B -> ( ( A R y <-> ( z F A ) R ( z F y ) ) <-> ( A R B <-> ( z F A ) R ( z F B ) ) ) )
16	11 15	sylan9bb	|- ( ( x = A /\ y = B ) -> ( ( x R y <-> ( z F x ) R ( z F y ) ) <-> ( A R B <-> ( z F A ) R ( z F B ) ) ) )
17	16	imbi2d	|- ( ( x = A /\ y = B ) -> ( ( z e. S -> ( x R y <-> ( z F x ) R ( z F y ) ) ) <-> ( z e. S -> ( A R B <-> ( z F A ) R ( z F B ) ) ) ) )
18	1 2 17 3	vtocl2	|- ( z e. S -> ( A R B <-> ( z F A ) R ( z F B ) ) )
19	7 18	vtoclga	|- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) )