Theorem caovdig 6489

Description: Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.)

Hypothesis

Ref	Expression
caovdig.1

Assertion

Ref	Expression
caovdig

Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,   ,S,,

Proof of Theorem caovdig

Step	Hyp	Ref	Expression
1		caovdig.1	. . 3
2	1	ralrimivvva 2879	. 2
3		oveq1 6303	. . . 4
4		oveq1 6303	. . . . 5
5		oveq1 6303	. . . . 5
6	4, 5	oveq12d 6314	. . . 4
7	3, 6	eqeq12d 2479	. . 3
8		oveq1 6303	. . . . 5
9	8	oveq2d 6312	. . . 4
10		oveq2 6304	. . . . 5
11	10	oveq1d 6311	. . . 4
12	9, 11	eqeq12d 2479	. . 3
13		oveq2 6304	. . . . 5
14	13	oveq2d 6312	. . . 4
15		oveq2 6304	. . . . 5
16	15	oveq2d 6312	. . . 4
17	14, 16	eqeq12d 2479	. . 3
18	7, 12, 17	rspc3v 3222	. 2
19	2, 18	mpan9 469	1

Syntax hints:  ->wi 4  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  A.wral 2807  (class class class)co 6296

This theorem is referenced by:  caovdid  6490  caovdi  6494  srgi  17163  ringi  17211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-iota 5556  df-fv 5601  df-ov 6299