Theorem caovordig 5998

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 2547	. 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 3979	. . . 4 |- ( x = A -> ( x R y <-> A R y ) )
4		oveq2 5844	. . . . 5 |- ( x = A -> ( z F x ) = ( z F A ) )
5	4	breq1d 3986	. . . 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 3980	. . . 4 |- ( y = B -> ( A R y <-> A R B ) )
8		oveq2 5844	. . . . 5 |- ( y = B -> ( z F y ) = ( z F B ) )
9	8	breq2d 3988	. . . 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 5843	. . . . 5 |- ( z = C -> ( z F A ) = ( C F A ) )
12		oveq1 5843	. . . . 5 |- ( z = C -> ( z F B ) = ( C F B ) )
13	11, 12	breq12d 3989	. . . 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 2841	. 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   /\ w3a 967   = wceq 1342   e. wcel 2135   A.wral 2442   class class class wbr 3976  (class class class)co 5836

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-un 3115  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-iota 5147  df-fv 5190  df-ov 5839

This theorem is referenced by:  caovordid  5999