Theorem caovdir2d 6491

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

Hypotheses

Ref	Expression
caovdir2d.1
caovdir2d.2
caovdir2d.3
caovdir2d.4
caovdir2d.cl
caovdir2d.com

Assertion

Ref	Expression
caovdir2d

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

Proof of Theorem caovdir2d

Step	Hyp	Ref	Expression
1		caovdir2d.1	. . 3
2		caovdir2d.4	. . 3
3		caovdir2d.2	. . 3
4		caovdir2d.3	. . 3
5	1, 2, 3, 4	caovdid 6490	. 2
6		caovdir2d.com	. . 3
7		caovdir2d.cl	. . . 4
8	7, 3, 4	caovcld 6468	. . 3
9	6, 8, 2	caovcomd 6471	. 2
10	6, 3, 2	caovcomd 6471	. . 3
11	6, 4, 2	caovcomd 6471	. . 3
12	10, 11	oveq12d 6314	. 2
13	5, 9, 12	3eqtr4d 2508	1

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

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