Theorem caovord 6845

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 6657	. . . 4 |- ( z = C -> ( z F A ) = ( C F A ) )
2		oveq1 6657	. . . 4 |- ( z = C -> ( z F B ) = ( C F B ) )
3	1, 2	breq12d 4666	. . 3 |- ( z = C -> ( ( z F A ) R ( z F B ) <-> ( C F A ) R ( C F B ) ) )
4	3	bibi2d 332	. 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 4656	. . . . . 6 |- ( x = A -> ( x R y <-> A R y ) )
8		oveq2 6658	. . . . . . 7 |- ( x = A -> ( z F x ) = ( z F A ) )
9	8	breq1d 4663	. . . . . 6 |- ( x = A -> ( ( z F x ) R ( z F y ) <-> ( z F A ) R ( z F y ) ) )
10	7, 9	bibi12d 335	. . . . 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 4657	. . . . . 6 |- ( y = B -> ( A R y <-> A R B ) )
12		oveq2 6658	. . . . . . 7 |- ( y = B -> ( z F y ) = ( z F B ) )
13	12	breq2d 4665	. . . . . 6 |- ( y = B -> ( ( z F A ) R ( z F y ) <-> ( z F A ) R ( z F B ) ) )
14	11, 13	bibi12d 335	. . . . 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 736	. . . 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 330	. . 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 3261	. 2 |- ( z e. S -> ( A R B <-> ( z F A ) R ( z F B ) ) )
19	4, 18	vtoclga 3272	1 |- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  caovord2  6846  caovord3  6847  genpcl  9830