Theorem undif4 3607

Description: Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem undif4

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm2.621 397	. . . . . . 7 |- ((x e. A -> -. x e. C) -> ((x e. A \/ -. x e. C) -> -. x e. C))
2		olc 373	. . . . . . 7 |- (-. x e. C -> (x e. A \/ -. x e. C))
3	1, 2	impbid1 194	. . . . . 6 |- ((x e. A -> -. x e. C) -> ((x e. A \/ -. x e. C) <-> -. x e. C))
4	3	anbi2d 684	. . . . 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)))
5		eldif 3221	. . . . . . 7 |- (x e. (B \ C) <-> (x e. B /\ -. x e. C))
6	5	orbi2i 505	. . . . . 6 |- ((x e. A \/ x e. (B \ C)) <-> (x e. A \/ (x e. B /\ -. x e. C)))
7		ordi 834	. . . . . 6 |- ((x e. A \/ (x e. B /\ -. x e. C)) <-> ((x e. A \/ x e. B) /\ (x e. A \/ -. x e. C)))
8	6, 7	bitri 240	. . . . 5 |- ((x e. A \/ x e. (B \ C)) <-> ((x e. A \/ x e. B) /\ (x e. A \/ -. x e. C)))
9		elun 3220	. . . . . 6 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
10	9	anbi1i 676	. . . . 5 |- ((x e. (A u. B) /\ -. x e. C) <-> ((x e. A \/ x e. B) /\ -. x e. C))
11	4, 8, 10	3bitr4g 279	. . . 4 |- ((x e. A -> -. x e. C) -> ((x e. A \/ x e. (B \ C)) <-> (x e. (A u. B) /\ -. x e. C)))
12		elun 3220	. . . 4 |- (x e. (A u. (B \ C)) <-> (x e. A \/ x e. (B \ C)))
13		eldif 3221	. . . 4 |- (x e. ((A u. B) \ C) <-> (x e. (A u. B) /\ -. x e. C))
14	11, 12, 13	3bitr4g 279	. . 3 |- ((x e. A -> -. x e. C) -> (x e. (A u. (B \ C)) <-> x e. ((A u. B) \ C)))
15	14	alimi 1559	. 2 |- (A.x(x e. A -> -. x e. C) -> A.x(x e. (A u. (B \ C)) <-> x e. ((A u. B) \ C)))
16		disj1 3593	. 2 |- ((A i^i C) = (/) <-> A.x(x e. A -> -. x e. C))
17		dfcleq 2347	. 2 |- ((A u. (B \ C)) = ((A u. B) \ C) <-> A.x(x e. (A u. (B \ C)) <-> x e. ((A u. B) \ C)))
18	15, 16, 17	3imtr4i 257	1 |- ((A i^i C) = (/) -> (A u. (B \ C)) = ((A u. B) \ C))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710   \ cdif 3206   u. cun 3207   i^i cin 3208  (/)c0 3550

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 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-nul 3551

This theorem is referenced by: (None)