Theorem caovord 6090

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 5925	. . . 4 |- ( z = C -> ( z F A ) = ( C F A ) )
2		oveq1 5925	. . . 4 |- ( z = C -> ( z F B ) = ( C F B ) )
3	1, 2	breq12d 4042	. . 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 4032	. . . . . 6 |- ( x = A -> ( x R y <-> A R y ) )
8		oveq2 5926	. . . . . . 7 |- ( x = A -> ( z F x ) = ( z F A ) )
9	8	breq1d 4039	. . . . . 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 4033	. . . . . 6 |- ( y = B -> ( A R y <-> A R B ) )
12		oveq2 5926	. . . . . . 7 |- ( y = B -> ( z F y ) = ( z F B ) )
13	12	breq2d 4041	. . . . . 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 2815	. 2 |- ( z e. S -> ( A R B <-> ( z F A ) R ( z F B ) ) )
19	4, 18	vtoclga 2826	1 |- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   _Vcvv 2760   class class class wbr 4029  (class class class)co 5918

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921

This theorem is referenced by:  caovord2  6091  caovord3  6092