Theorem caovdird 5699

Description: Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Hypotheses

Ref	Expression
caovdirg.1	|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. K ) ) -> ( ( x F y ) G z ) = ( ( x G z ) H ( y G z ) ) )
caovdird.2	|- ( ph -> A e. S )
caovdird.3	|- ( ph -> B e. S )
caovdird.4	|- ( ph -> C e. K )

Assertion

Ref	Expression
caovdird	|- ( ph -> ( ( A F B ) G C ) = ( ( A G C ) H ( B G C ) ) )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   ph, x, y, z   x, F, y, z   x, G, y, z   x, H, y, z   x, K, y, z   x, S, y, z

Proof of Theorem caovdird

Step	Hyp	Ref	Expression
1		id 19	. 2 |- ( ph -> ph )
2		caovdird.2	. 2 |- ( ph -> A e. S )
3		caovdird.3	. 2 |- ( ph -> B e. S )
4		caovdird.4	. 2 |- ( ph -> C e. K )
5		caovdirg.1	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. K ) ) -> ( ( x F y ) G z ) = ( ( x G z ) H ( y G z ) ) )
6	5	caovdirg 5698	. 2 |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. K ) ) -> ( ( A F B ) G C ) = ( ( A G C ) H ( B G C ) ) )
7	1, 2, 3, 4, 6	syl13anc 1171	1 |- ( ph -> ( ( A F B ) G C ) = ( ( A G C ) H ( B G C ) ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   = wceq 1284   e. wcel 1433  (class class class)co 5532

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535

This theorem is referenced by:  caovdilemd  5712  recexgt0sr  6950