Theorem caovdid 6122

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 6121	. 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 1252	1 |- ( ph -> ( A G ( B F C ) ) = ( ( A G B ) H ( A G C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2176  (class class class)co 5944

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279  df-ov 5947

This theorem is referenced by:  caovdir2d  6123  caovlem2d  6139  caofdig  6192  ltanqg  7513