Theorem caovord 5692

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

Hypotheses

Ref	Expression
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

Ref	Expression
caovord	|- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   x, F, y, z   x, R, y, z   x, S, y, z

Proof of Theorem caovord

Step	Hyp	Ref	Expression
1		oveq1 5539	. . . 4 |- ( z = C -> ( z F A ) = ( C F A ) )
2		oveq1 5539	. . . 4 |- ( z = C -> ( z F B ) = ( C F B ) )
3	1, 2	breq12d 3798	. . 3 |- ( z = C -> ( ( z F A ) R ( z F B ) <-> ( C F A ) R ( C F B ) ) )
4	3	bibi2d 230	. 2 |- ( z = C -> ( ( A R B <-> ( z F A ) R ( z F B ) ) <-> ( A R B <-> ( C F A ) R ( C F B ) ) ) )
5		caovord.1	. . 3 |- A e. _V
6		caovord.2	. . 3 |- B e. _V
7		breq1 3788	. . . . . 6 |- ( x = A -> ( x R y <-> A R y ) )
8		oveq2 5540	. . . . . . 7 |- ( x = A -> ( z F x ) = ( z F A ) )
9	8	breq1d 3795	. . . . . 6 |- ( x = A -> ( ( z F x ) R ( z F y ) <-> ( z F A ) R ( z F y ) ) )
10	7, 9	bibi12d 233	. . . . 5 |- ( x = A -> ( ( x R y <-> ( z F x ) R ( z F y ) ) <-> ( A R y <-> ( z F A ) R ( z F y ) ) ) )
11		breq2 3789	. . . . . 6 |- ( y = B -> ( A R y <-> A R B ) )
12		oveq2 5540	. . . . . . 7 |- ( y = B -> ( z F y ) = ( z F B ) )
13	12	breq2d 3797	. . . . . 6 |- ( y = B -> ( ( z F A ) R ( z F y ) <-> ( z F A ) R ( z F B ) ) )
14	11, 13	bibi12d 233	. . . . 5 |- ( y = B -> ( ( A R y <-> ( z F A ) R ( z F y ) ) <-> ( A R B <-> ( z F A ) R ( z F B ) ) ) )
15	10, 14	sylan9bb 449	. . . 4 |- ( ( x = A /\ y = B ) -> ( ( x R y <-> ( z F x ) R ( z F y ) ) <-> ( A R B <-> ( z F A ) R ( z F B ) ) ) )
16	15	imbi2d 228	. . 3 |- ( ( 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 ) ) ) ) )
17		caovord.3	. . 3 |- ( z e. S -> ( x R y <-> ( z F x ) R ( z F y ) ) )
18	5, 6, 16, 17	vtocl2 2654	. 2 |- ( z e. S -> ( A R B <-> ( z F A ) R ( z F B ) ) )
19	4, 18	vtoclga 2664	1 |- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   _Vcvv 2601   class class class wbr 3785  (class class class)co 5532

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535

This theorem is referenced by:  caovord2  5693  caovord3  5694