Theorem difdifdir 2358

Description: Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16.

Assertion

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

Proof of Theorem difdifdir

Step	Hyp	Ref	Expression
1		difdisj 2349	. . . . 5 |- (C i^i (A \ C)) = (/)
2		incom 2219	. . . . 5 |- (C i^i (A \ C)) = ((A \ C) i^i C)
3	1, 2	eqtr3i 1504	. . . 4 |- (/) = ((A \ C) i^i C)
4	3	uneq2i 2192	. . 3 |- (((A \ C) i^i (V \ B)) u. (/)) = (((A \ C) i^i (V \ B)) u. ((A \ C) i^i C))
5		invdif 2260	. . . 4 |- ((A \ C) i^i (V \ B)) = ((A \ C) \ B)
6		un0 2309	. . . 4 |- (((A \ C) i^i (V \ B)) u. (/)) = ((A \ C) i^i (V \ B))
7		dif23 2275	. . . 4 |- ((A \ B) \ C) = ((A \ C) \ B)
8	5, 6, 7	3eqtr4ri 1513	. . 3 |- ((A \ B) \ C) = (((A \ C) i^i (V \ B)) u. (/))
9		indi 2262	. . 3 |- ((A \ C) i^i ((V \ B) u. C)) = (((A \ C) i^i (V \ B)) u. ((A \ C) i^i C))
10	4, 8, 9	3eqtr4i 1512	. 2 |- ((A \ B) \ C) = ((A \ C) i^i ((V \ B) u. C))
11		indm 2273	. . . 4 |- (V \ (B i^i (V \ C))) = ((V \ B) u. (V \ (V \ C)))
12		invdif 2260	. . . . 5 |- (B i^i (V \ C)) = (B \ C)
13	12	difeq2i 2167	. . . 4 |- (V \ (B i^i (V \ C))) = (V \ (B \ C))
14		ddif 2180	. . . . 5 |- (V \ (V \ C)) = C
15	14	uneq2i 2192	. . . 4 |- ((V \ B) u. (V \ (V \ C))) = ((V \ B) u. C)
16	11, 13, 15	3eqtr3ri 1511	. . 3 |- ((V \ B) u. C) = (V \ (B \ C))
17	16	ineq2i 2225	. 2 |- ((A \ C) i^i ((V \ B) u. C)) = ((A \ C) i^i (V \ (B \ C)))
18		invdif 2260	. 2 |- ((A \ C) i^i (V \ (B \ C))) = ((A \ C) \ (B \ C))
19	10, 17, 18	3eqtri 1506	1 |- ((A \ B) \ C) = ((A \ C) \ (B \ C))

Syntax hints:   = wceq 960  Vcvv 1818   \ cdif 2055   u. cun 2056   i^i cin 2057  (/)c0 2291

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1176  df-clab 1470  df-cleq 1475  df-clel 1478  df-ne 1594  df-v 1819  df-dif 2060  df-un 2061  df-in 2062  df-ss 2064  df-nul 2292

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