Theorem undif4 2296

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 2028	. . . . . . 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 2144	. . . . . 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 2144	. . . 4 |- (x e. (A u. (B \ C)) <-> (x e. A \/ x e. (B \ C)))
15		eldif 2028	. . . 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 968	. 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 2283	. 2 |- ((A i^i C) = (/) <-> A.x(x e. A -> -. x e. C))
19		dfcleq 1447	. 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 950   = wceq 1099   e. wcel 1105   \ cdif 2015   u. cun 2016   i^i cin 2017  (/)c0 2251

This theorem is referenced by:  phplem1 4440

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-ral 1625  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-nul 2252

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