Theorem caovordig 6084

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 2577	. 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 4032	. . . 4 |- ( x = A -> ( x R y <-> A R y ) )
4		oveq2 5926	. . . . 5 |- ( x = A -> ( z F x ) = ( z F A ) )
5	4	breq1d 4039	. . . 4 |- ( x = A -> ( ( z F x ) R ( z F y ) <-> ( z F A ) R ( z F y ) ) )
6	3, 5	imbi12d 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 4033	. . . 4 |- ( y = B -> ( A R y <-> A R B ) )
8		oveq2 5926	. . . . 5 |- ( y = B -> ( z F y ) = ( z F B ) )
9	8	breq2d 4041	. . . 4 |- ( y = B -> ( ( z F A ) R ( z F y ) <-> ( z F A ) R ( z F B ) ) )
10	7, 9	imbi12d 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 5925	. . . . 5 |- ( z = C -> ( z F A ) = ( C F A ) )
12		oveq1 5925	. . . . 5 |- ( z = C -> ( z F B ) = ( C F B ) )
13	11, 12	breq12d 4042	. . . 4 |- ( z = C -> ( ( z F A ) R ( z F B ) <-> ( C F A ) R ( C F B ) ) )
14	13	imbi2d 230	. . 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 2880	. 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 281	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 104   /\ w3a 980   = wceq 1364   e. wcel 2164   A.wral 2472   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-ral 2477  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:  caovordid  6085