Theorem caovordid 5945

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

Hypotheses

Ref	Expression
caovordig.1	|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y -> ( z F x ) R ( z F y ) ) )
caovordid.2	|- ( ph -> A e. S )
caovordid.3	|- ( ph -> B e. S )
caovordid.4	|- ( ph -> C e. S )

Assertion

Ref	Expression
caovordid	|- ( ph -> ( 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 caovordid

Step	Hyp	Ref	Expression
1		id 19	. 2 |- ( ph -> ph )
2		caovordid.2	. 2 |- ( ph -> A e. S )
3		caovordid.3	. 2 |- ( ph -> B e. S )
4		caovordid.4	. 2 |- ( ph -> C e. S )
5		caovordig.1	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y -> ( z F x ) R ( z F y ) ) )
6	5	caovordig 5944	. 2 |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) -> ( A R B -> ( C F A ) R ( C F B ) ) )
7	1, 2, 3, 4, 6	syl13anc 1219	1 |- ( ph -> ( A R B -> ( C F A ) R ( C F B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   e. wcel 1481   class class class wbr 3937  (class class class)co 5782

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fv 5139  df-ov 5785

This theorem is referenced by: (None)