Theorem caovdird 6211

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 6210	. 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 1276	1 |- ( ph -> ( ( A F B ) G C ) = ( ( A G C ) H ( B G C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2202  (class class class)co 6028

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-ov 6031

This theorem is referenced by:  caovdilemd  6224  recexgt0sr  8053