Theorem undif4 1744

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		pm2.61 109	. . . . . . . 8 |- ((x e. A -> -. x e. C) -> ((-. x e. A -> -. x e. C) -> -. x e. C))
2		ax-1 3	. . . . . . . . 9 |- (-. x e. C -> (-. x e. A -> -. x e. C))
3	2	a1i 7	. . . . . . . 8 |- ((x e. A -> -. x e. C) -> (-. x e. C -> (-. x e. A -> -. x e. C)))
4	1, 3	impbid 397	. . . . . . 7 |- ((x e. A -> -. x e. C) -> ((-. x e. A -> -. x e. C) <-> -. x e. C))
5		df-or 197	. . . . . . 7 |- ((x e. A \/ -. x e. C) <-> (-. x e. A -> -. x e. C))
6	4, 5	syl5bb 410	. . . . . 6 |- ((x e. A -> -. x e. C) -> ((x e. A \/ -. x e. C) <-> -. x e. C))
7	6	anbi2d 468	. . . . 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)))
8		eldif 1496	. . . . . . 7 |- (x e. (B \ C) <-> (x e. B /\ -. x e. C))
9	8	orbi2i 214	. . . . . 6 |- ((x e. A \/ x e. (B \ C)) <-> (x e. A \/ (x e. B /\ -. x e. C)))
10		ordi 452	. . . . . 6 |- ((x e. A \/ (x e. B /\ -. x e. C)) <-> ((x e. A \/ x e. B) /\ (x e. A \/ -. x e. C)))
11	9, 10	bitr 151	. . . . 5 |- ((x e. A \/ x e. (B \ C)) <-> ((x e. A \/ x e. B) /\ (x e. A \/ -. x e. C)))
12		elun 1601	. . . . . 6 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
13	12	anbi1i 368	. . . . 5 |- ((x e. (A u. B) /\ -. x e. C) <-> ((x e. A \/ x e. B) /\ -. x e. C))
14	7, 11, 13	3bitr4g 428	. . . 4 |- ((x e. A -> -. x e. C) -> ((x e. A \/ x e. (B \ C)) <-> (x e. (A u. B) /\ -. x e. C)))
15		elun 1601	. . . 4 |- (x e. (A u. (B \ C)) <-> (x e. A \/ x e. (B \ C)))
16		eldif 1496	. . . 4 |- (x e. ((A u. B) \ C) <-> (x e. (A u. B) /\ -. x e. C))
17	14, 15, 16	3bitr4g 428	. . 3 |- ((x e. A -> -. x e. C) -> (x e. (A u. (B \ C)) <-> x e. ((A u. B) \ C)))
18	17	19.20i 691	. 2 |- (A.x(x e. A -> -. x e. C) -> A.x(x e. (A u. (B \ C)) <-> x e. ((A u. B) \ C)))
19		disj1 1734	. 2 |- ((A i^i C) = (/) <-> A.x(x e. A -> -. x e. C))
20		dfcleq 1098	. 2 |- ((A u. (B \ C)) = ((A u. B) \ C) <-> A.x(x e. (A u. (B \ C)) <-> x e. ((A u. B) \ C)))
21	18, 19, 20	3imtr4 192	1 |- ((A i^i C) = (/) -> (A u. (B \ C)) = ((A u. B) \ C))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196  A.wal 672   = wceq 1091   e. wcel 1092   \ cdif 1484   u. cun 1485   i^i cin 1486  (/)c0 1707

This theorem is referenced by:  phplem2 3404

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-nul 1708

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