Theorem caovdird 6076

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 6075	. 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 1251	1 |- ( ph -> ( ( A F B ) G C ) = ( ( A G C ) H ( B G C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2160  (class class class)co 5897

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 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5900

This theorem is referenced by:  caovdilemd  6089  recexgt0sr  7803