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Mirrors > Home > ILE Home > Th. List > indifdir | Unicode version |
Description: Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.) |
Ref | Expression |
---|---|
indifdir |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3300 | . . . 4 | |
2 | elin 3300 | . . . . 5 | |
3 | 2 | notbii 658 | . . . 4 |
4 | 1, 3 | anbi12i 456 | . . 3 |
5 | eldif 3120 | . . 3 | |
6 | elin 3300 | . . . . 5 | |
7 | eldif 3120 | . . . . . 6 | |
8 | 7 | anbi1i 454 | . . . . 5 |
9 | 6, 8 | bitri 183 | . . . 4 |
10 | an32 552 | . . . . 5 | |
11 | simpl 108 | . . . . . . . 8 | |
12 | 11 | con3i 622 | . . . . . . 7 |
13 | 12 | anim2i 340 | . . . . . 6 |
14 | simpl 108 | . . . . . . 7 | |
15 | ax-in2 605 | . . . . . . . . . . 11 | |
16 | 15 | expcomd 1428 | . . . . . . . . . 10 |
17 | 16 | impcom 124 | . . . . . . . . 9 |
18 | dfnot 1360 | . . . . . . . . 9 | |
19 | 17, 18 | sylibr 133 | . . . . . . . 8 |
20 | 19 | adantll 468 | . . . . . . 7 |
21 | 14, 20 | jca 304 | . . . . . 6 |
22 | 13, 21 | impbii 125 | . . . . 5 |
23 | 10, 22 | bitri 183 | . . . 4 |
24 | 9, 23 | bitri 183 | . . 3 |
25 | 4, 5, 24 | 3bitr4ri 212 | . 2 |
26 | 25 | eqriv 2161 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1342 wfal 1347 wcel 2135 cdif 3108 cin 3110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-dif 3113 df-in 3117 |
This theorem is referenced by: (None) |
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