Theorem caovordg 6020

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 2553	. 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 3992	. . . 4 |- ( x = A -> ( x R y <-> A R y ) )
4		oveq2 5861	. . . . 5 |- ( x = A -> ( z F x ) = ( z F A ) )
5	4	breq1d 3999	. . . 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 3993	. . . 4 |- ( y = B -> ( A R y <-> A R B ) )
8		oveq2 5861	. . . . 5 |- ( y = B -> ( z F y ) = ( z F B ) )
9	8	breq2d 4001	. . . 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 5860	. . . . 5 |- ( z = C -> ( z F A ) = ( C F A ) )
12		oveq1 5860	. . . . 5 |- ( z = C -> ( z F B ) = ( C F B ) )
13	11, 12	breq12d 4002	. . . 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 2850	. 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 973   = wceq 1348   e. wcel 2141   A.wral 2448   class class class wbr 3989  (class class class)co 5853

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by:  caovordd  6021