Theorem caovdid 5676

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

Hypotheses

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

Assertion

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

Step	Hyp	Ref	Expression
1		id 19	. 2 |- ( ph -> ph )
2		caovdid.2	. 2 |- ( ph -> A e. K )
3		caovdid.3	. 2 |- ( ph -> B e. S )
4		caovdid.4	. 2 |- ( ph -> C e. S )
5		caovdig.1	. . 3 |- ( ( ph /\ ( x e. K /\ y e. S /\ z e. S ) ) -> ( x G ( y F z ) ) = ( ( x G y ) H ( x G z ) ) )
6	5	caovdig 5675	. 2 |- ( ( ph /\ ( A e. K /\ B e. S /\ C e. S ) ) -> ( A G ( B F C ) ) = ( ( A G B ) H ( A G C ) ) )
7	1, 2, 3, 4, 6	syl13anc 1137	1 |- ( ph -> ( A G ( B F C ) ) = ( ( A G B ) H ( A G C ) ) )

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885   = wceq 1243   e. wcel 1393  (class class class)co 5512

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by:  caovdir2d  5677  caovlem2d  5693  ltanqg  6498