Theorem caovord 6204

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 6035	. . . 4 |- ( z = C -> ( z F A ) = ( C F A ) )
2		oveq1 6035	. . . 4 |- ( z = C -> ( z F B ) = ( C F B ) )
3	1, 2	breq12d 4106	. . 3 |- ( z = C -> ( ( z F A ) R ( z F B ) <-> ( C F A ) R ( C F B ) ) )
4	3	bibi2d 232	. 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 4096	. . . . . 6 |- ( x = A -> ( x R y <-> A R y ) )
8		oveq2 6036	. . . . . . 7 |- ( x = A -> ( z F x ) = ( z F A ) )
9	8	breq1d 4103	. . . . . 6 |- ( x = A -> ( ( z F x ) R ( z F y ) <-> ( z F A ) R ( z F y ) ) )
10	7, 9	bibi12d 235	. . . . 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 4097	. . . . . 6 |- ( y = B -> ( A R y <-> A R B ) )
12		oveq2 6036	. . . . . . 7 |- ( y = B -> ( z F y ) = ( z F B ) )
13	12	breq2d 4105	. . . . . 6 |- ( y = B -> ( ( z F A ) R ( z F y ) <-> ( z F A ) R ( z F B ) ) )
14	11, 13	bibi12d 235	. . . . 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 462	. . . 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 230	. . 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 2860	. 2 |- ( z e. S -> ( A R B <-> ( z F A ) R ( z F B ) ) )
19	4, 18	vtoclga 2871	1 |- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   _Vcvv 2803   class class class wbr 4093  (class class class)co 6028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-ov 6031

This theorem is referenced by:  caovord2  6205  caovord3  6206