Theorem difdifdir 3638

Description: Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16. (Contributed by NM, 18-Aug-2004.)

Assertion

Ref	Expression
difdifdir	|- ((A \ B) \ C) = ((A \ C) \ (B \ C))

Proof of Theorem difdifdir

Step	Hyp	Ref	Expression
1		dif32 3518	. . . . 5 |- ((A \ B) \ C) = ((A \ C) \ B)
2		invdif 3497	. . . . 5 |- ((A \ C) i^i (_V \ B)) = ((A \ C) \ B)
3	1, 2	eqtr4i 2376	. . . 4 |- ((A \ B) \ C) = ((A \ C) i^i (_V \ B))
4		un0 3576	. . . 4 |- (((A \ C) i^i (_V \ B)) u. (/)) = ((A \ C) i^i (_V \ B))
5	3, 4	eqtr4i 2376	. . 3 |- ((A \ B) \ C) = (((A \ C) i^i (_V \ B)) u. (/))
6		indi 3502	. . . 4 |- ((A \ C) i^i ((_V \ B) u. C)) = (((A \ C) i^i (_V \ B)) u. ((A \ C) i^i C))
7		disjdif 3623	. . . . . 6 |- (C i^i (A \ C)) = (/)
8		incom 3449	. . . . . 6 |- (C i^i (A \ C)) = ((A \ C) i^i C)
9	7, 8	eqtr3i 2375	. . . . 5 |- (/) = ((A \ C) i^i C)
10	9	uneq2i 3416	. . . 4 |- (((A \ C) i^i (_V \ B)) u. (/)) = (((A \ C) i^i (_V \ B)) u. ((A \ C) i^i C))
11	6, 10	eqtr4i 2376	. . 3 |- ((A \ C) i^i ((_V \ B) u. C)) = (((A \ C) i^i (_V \ B)) u. (/))
12	5, 11	eqtr4i 2376	. 2 |- ((A \ B) \ C) = ((A \ C) i^i ((_V \ B) u. C))
13		ddif 3399	. . . . 5 |- (_V \ (_V \ C)) = C
14	13	uneq2i 3416	. . . 4 |- ((_V \ B) u. (_V \ (_V \ C))) = ((_V \ B) u. C)
15		indm 3514	. . . . 5 |- (_V \ (B i^i (_V \ C))) = ((_V \ B) u. (_V \ (_V \ C)))
16		invdif 3497	. . . . . 6 |- (B i^i (_V \ C)) = (B \ C)
17	16	difeq2i 3383	. . . . 5 |- (_V \ (B i^i (_V \ C))) = (_V \ (B \ C))
18	15, 17	eqtr3i 2375	. . . 4 |- ((_V \ B) u. (_V \ (_V \ C))) = (_V \ (B \ C))
19	14, 18	eqtr3i 2375	. . 3 |- ((_V \ B) u. C) = (_V \ (B \ C))
20	19	ineq2i 3455	. 2 |- ((A \ C) i^i ((_V \ B) u. C)) = ((A \ C) i^i (_V \ (B \ C)))
21		invdif 3497	. 2 |- ((A \ C) i^i (_V \ (B \ C))) = ((A \ C) \ (B \ C))
22	12, 20, 21	3eqtri 2377	1 |- ((A \ B) \ C) = ((A \ C) \ (B \ C))

Syntax hints:   = wceq 1642  _Vcvv 2860   \ cdif 3207   u. cun 3208   i^i cin 3209  (/)c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by: (None)