Theorem undif4 2315

Description: Distribute union over difference.

Assertion

Ref	Expression
undif4	|- ((A i^i C) = (/) -> (A u. (B \ C)) = ((A u. B) \ C))

Proof of Theorem undif4

Step	Hyp	Ref	Expression
1		pm2.61 124	. . . . . . . 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 515	. . . . . . 7 |- ((x e. A -> -. x e. C) -> ((-. x e. A -> -. x e. C) <-> -. x e. C))
4		df-or 224	. . . . . . 7 |- ((x e. A \/ -. x e. C) <-> (-. x e. A -> -. x e. C))
5	3, 4	syl5bb 530	. . . . . 6 |- ((x e. A -> -. x e. C) -> ((x e. A \/ -. x e. C) <-> -. x e. C))
6	5	anbi2d 614	. . . . 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 2047	. . . . . . 7 |- (x e. (B \ C) <-> (x e. B /\ -. x e. C))
8	7	orbi2i 255	. . . . . 6 |- ((x e. A \/ x e. (B \ C)) <-> (x e. A \/ (x e. B /\ -. x e. C)))
9		ordi 594	. . . . . 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	bitr 173	. . . . 5 |- ((x e. A \/ x e. (B \ C)) <-> ((x e. A \/ x e. B) /\ (x e. A \/ -. x e. C)))
11		elun 2163	. . . . . 6 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
12	11	anbi1i 480	. . . . 5 |- ((x e. (A u. B) /\ -. x e. C) <-> ((x e. A \/ x e. B) /\ -. x e. C))
13	6, 10, 12	3bitr4g 553	. . . 4 |- ((x e. A -> -. x e. C) -> ((x e. A \/ x e. (B \ C)) <-> (x e. (A u. B) /\ -. x e. C)))
14		elun 2163	. . . 4 |- (x e. (A u. (B \ C)) <-> (x e. A \/ x e. (B \ C)))
15		eldif 2047	. . . 4 |- (x e. ((A u. B) \ C) <-> (x e. (A u. B) /\ -. x e. C))
16	13, 14, 15	3bitr4g 553	. . 3 |- ((x e. A -> -. x e. C) -> (x e. (A u. (B \ C)) <-> x e. ((A u. B) \ C)))
17	16	19.20i 989	. 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 2302	. 2 |- ((A i^i C) = (/) <-> A.x(x e. A -> -. x e. C))
19		dfcleq 1463	. 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	3imtr4 219	1 |- ((A i^i C) = (/) -> (A u. (B \ C)) = ((A u. B) \ C))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955   \ cdif 2034   u. cun 2035   i^i cin 2036  (/)c0 2270

This theorem is referenced by:  phplem1 4488

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-nul 2271

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