Theorem undif4OLD 2931

Description: Distribute union over difference.

Assertion

Ref	Expression
undif4OLD	|- ((A i^i C) = (/) -> (A u. (B \ C)) = ((A u. B) \ C))

Proof of Theorem undif4OLD

Step	Hyp	Ref	Expression
1		pm2.61 139	. . . . . . . 8 |- ((x e. A -> -. x e. C) -> ((-. x e. A -> -. x e. C) -> -. x e. C))
2		ax-1 4	. . . . . . . 8 |- (-. x e. C -> (-. x e. A -> -. x e. C))
3	1, 2	impbid1 575	. . . . . . 7 |- ((x e. A -> -. x e. C) -> ((-. x e. A -> -. x e. C) <-> -. x e. C))
4		df-or 241	. . . . . . 7 |- ((x e. A \/ -. x e. C) <-> (-. x e. A -> -. x e. C))
5	3, 4	syl5bb 591	. . . . . 6 |- ((x e. A -> -. x e. C) -> ((x e. A \/ -. x e. C) <-> -. x e. C))
6	5	anbi2d 678	. . . . 5 |- ((x e. A -> -. x e. C) -> (((x e. A \/ x e. B) /\ (x e. A \/ -. x e. C)) <-> ((x e. A \/ x e. B) /\ -. x e. C)))
7		eldif 2609	. . . . . . 7 |- (x e. (B \ C) <-> (x e. B /\ -. x e. C))
8	7	orbi2i 275	. . . . . 6 |- ((x e. A \/ x e. (B \ C)) <-> (x e. A \/ (x e. B /\ -. x e. C)))
9		ordi 656	. . . . . 6 |- ((x e. A \/ (x e. B /\ -. x e. C)) <-> ((x e. A \/ x e. B) /\ (x e. A \/ -. x e. C)))
10	8, 9	bitri 190	. . . . 5 |- ((x e. A \/ x e. (B \ C)) <-> ((x e. A \/ x e. B) /\ (x e. A \/ -. x e. C)))
11		elun 2741	. . . . . 6 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
12	11	anbi1i 539	. . . . 5 |- ((x e. (A u. B) /\ -. x e. C) <-> ((x e. A \/ x e. B) /\ -. x e. C))
13	6, 10, 12	3bitr4g 614	. . . 4 |- ((x e. A -> -. x e. C) -> ((x e. A \/ x e. (B \ C)) <-> (x e. (A u. B) /\ -. x e. C)))
14		elun 2741	. . . 4 |- (x e. (A u. (B \ C)) <-> (x e. A \/ x e. (B \ C)))
15		eldif 2609	. . . 4 |- (x e. ((A u. B) \ C) <-> (x e. (A u. B) /\ -. x e. C))
16	13, 14, 15	3bitr4g 614	. . 3 |- ((x e. A -> -. x e. C) -> (x e. (A u. (B \ C)) <-> x e. ((A u. B) \ C)))
17	16	alimi 1338	. 2 |- (A.x(x e. A -> -. x e. C) -> A.x(x e. (A u. (B \ C)) <-> x e. ((A u. B) \ C)))
18		disj1 2915	. 2 |- ((A i^i C) = (/) <-> A.x(x e. A -> -. x e. C))
19		dfcleq 1878	. 2 |- ((A u. (B \ C)) = ((A u. B) \ C) <-> A.x(x e. (A u. (B \ C)) <-> x e. ((A u. B) \ C)))
20	17, 18, 19	3imtr4i 236	1 |- ((A i^i C) = (/) -> (A u. (B \ C)) = ((A u. B) \ C))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300   \ cdif 2590   u. cun 2591   i^i cin 2592  (/)c0 2875

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-nul 2876

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