Theorem caovord 5935

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 5774	. . . 4 |- ( z = C -> ( z F A ) = ( C F A ) )
2		oveq1 5774	. . . 4 |- ( z = C -> ( z F B ) = ( C F B ) )
3	1, 2	breq12d 3937	. . 3 |- ( z = C -> ( ( z F A ) R ( z F B ) <-> ( C F A ) R ( C F B ) ) )
4	3	bibi2d 231	. 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 3927	. . . . . 6 |- ( x = A -> ( x R y <-> A R y ) )
8		oveq2 5775	. . . . . . 7 |- ( x = A -> ( z F x ) = ( z F A ) )
9	8	breq1d 3934	. . . . . 6 |- ( x = A -> ( ( z F x ) R ( z F y ) <-> ( z F A ) R ( z F y ) ) )
10	7, 9	bibi12d 234	. . . . 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 3928	. . . . . 6 |- ( y = B -> ( A R y <-> A R B ) )
12		oveq2 5775	. . . . . . 7 |- ( y = B -> ( z F y ) = ( z F B ) )
13	12	breq2d 3936	. . . . . 6 |- ( y = B -> ( ( z F A ) R ( z F y ) <-> ( z F A ) R ( z F B ) ) )
14	11, 13	bibi12d 234	. . . . 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 457	. . . 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 229	. . 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 2736	. 2 |- ( z e. S -> ( A R B <-> ( z F A ) R ( z F B ) ) )
19	4, 18	vtoclga 2747	1 |- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   _Vcvv 2681   class class class wbr 3924  (class class class)co 5767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-iota 5083  df-fv 5126  df-ov 5770

This theorem is referenced by:  caovord2  5936  caovord3  5937