Theorem caovordig 5868

Description: Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)

Hypothesis

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

Assertion

Ref	Expression
caovordig	|- ( ( 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 caovordig

Step	Hyp	Ref	Expression
1		caovordig.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 2474	. 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 3878	. . . 4 |- ( x = A -> ( x R y <-> A R y ) )
4		oveq2 5714	. . . . 5 |- ( x = A -> ( z F x ) = ( z F A ) )
5	4	breq1d 3885	. . . 4 |- ( x = A -> ( ( z F x ) R ( z F y ) <-> ( z F A ) R ( z F y ) ) )
6	3, 5	imbi12d 233	. . 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 3879	. . . 4 |- ( y = B -> ( A R y <-> A R B ) )
8		oveq2 5714	. . . . 5 |- ( y = B -> ( z F y ) = ( z F B ) )
9	8	breq2d 3887	. . . 4 |- ( y = B -> ( ( z F A ) R ( z F y ) <-> ( z F A ) R ( z F B ) ) )
10	7, 9	imbi12d 233	. . 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 5713	. . . . 5 |- ( z = C -> ( z F A ) = ( C F A ) )
12		oveq1 5713	. . . . 5 |- ( z = C -> ( z F B ) = ( C F B ) )
13	11, 12	breq12d 3888	. . . 4 |- ( z = C -> ( ( z F A ) R ( z F B ) <-> ( C F A ) R ( C F B ) ) )
14	13	imbi2d 229	. . 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 2759	. 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 277	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   /\ w3a 930   = wceq 1299   e. wcel 1448   A.wral 2375   class class class wbr 3875  (class class class)co 5706

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-iota 5024  df-fv 5067  df-ov 5709

This theorem is referenced by:  caovordid  5869