Theorem caovordg 6009

Description: Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 30-Dec-2014.)

Hypothesis

Ref	Expression
caovordg.1	|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y <-> ( z F x ) R ( z F y ) ) )

Assertion

Ref	Expression
caovordg	|- ( ( ph /\ ( A e. S /\ B e. S /\ 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   ph, x, y, z   x, F, y, z   x, R, y, z   x, S, y, z

Proof of Theorem caovordg

Step	Hyp	Ref	Expression
1		caovordg.1	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y <-> ( z F x ) R ( z F y ) ) )
2	1	ralrimivvva 2549	. 2 |- ( ph -> A. x e. S A. y e. S A. z e. S ( x R y <-> ( z F x ) R ( z F y ) ) )
3		breq1 3985	. . . 4 |- ( x = A -> ( x R y <-> A R y ) )
4		oveq2 5850	. . . . 5 |- ( x = A -> ( z F x ) = ( z F A ) )
5	4	breq1d 3992	. . . 4 |- ( x = A -> ( ( z F x ) R ( z F y ) <-> ( z F A ) R ( z F y ) ) )
6	3, 5	bibi12d 234	. . 3 |- ( x = A -> ( ( x R y <-> ( z F x ) R ( z F y ) ) <-> ( A R y <-> ( z F A ) R ( z F y ) ) ) )
7		breq2 3986	. . . 4 |- ( y = B -> ( A R y <-> A R B ) )
8		oveq2 5850	. . . . 5 |- ( y = B -> ( z F y ) = ( z F B ) )
9	8	breq2d 3994	. . . 4 |- ( y = B -> ( ( z F A ) R ( z F y ) <-> ( z F A ) R ( z F B ) ) )
10	7, 9	bibi12d 234	. . 3 |- ( y = B -> ( ( A R y <-> ( z F A ) R ( z F y ) ) <-> ( A R B <-> ( z F A ) R ( z F B ) ) ) )
11		oveq1 5849	. . . . 5 |- ( z = C -> ( z F A ) = ( C F A ) )
12		oveq1 5849	. . . . 5 |- ( z = C -> ( z F B ) = ( C F B ) )
13	11, 12	breq12d 3995	. . . 4 |- ( z = C -> ( ( z F A ) R ( z F B ) <-> ( C F A ) R ( C F B ) ) )
14	13	bibi2d 231	. . 3 |- ( z = C -> ( ( A R B <-> ( z F A ) R ( z F B ) ) <-> ( A R B <-> ( C F A ) R ( C F B ) ) ) )
15	6, 10, 14	rspc3v 2846	. 2 |- ( ( A e. S /\ B e. S /\ C e. S ) -> ( A. x e. S A. y e. S A. z e. S ( x R y <-> ( z F x ) R ( z F y ) ) -> ( A R B <-> ( C F A ) R ( C F B ) ) ) )
16	2, 15	mpan9 279	1 |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) -> ( A R B <-> ( C F A ) R ( C F B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   A.wral 2444   class class class wbr 3982  (class class class)co 5842

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by:  caovordd  6010