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Mirrors > Home > ILE Home > Th. List > eueq3dc | Unicode version |
Description: Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.) |
Ref | Expression |
---|---|
eueq3dc.1 | |
eueq3dc.2 | |
eueq3dc.3 | |
eueq3dc.4 |
Ref | Expression |
---|---|
eueq3dc | DECID DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dcor 843 | . 2 DECID DECID DECID | |
2 | df-dc 743 | . . 3 DECID | |
3 | eueq3dc.1 | . . . . . . 7 | |
4 | 3 | eueq1 2713 | . . . . . 6 |
5 | ibar 285 | . . . . . . . . 9 | |
6 | pm2.45 657 | . . . . . . . . . . . . 13 | |
7 | eueq3dc.4 | . . . . . . . . . . . . . . 15 | |
8 | 7 | imnani 625 | . . . . . . . . . . . . . 14 |
9 | 8 | con2i 557 | . . . . . . . . . . . . 13 |
10 | 6, 9 | jaoi 636 | . . . . . . . . . . . 12 |
11 | 10 | con2i 557 | . . . . . . . . . . 11 |
12 | 6 | con2i 557 | . . . . . . . . . . . . 13 |
13 | 12 | bianfd 855 | . . . . . . . . . . . 12 |
14 | 8 | bianfd 855 | . . . . . . . . . . . 12 |
15 | 13, 14 | orbi12d 707 | . . . . . . . . . . 11 |
16 | 11, 15 | mtbid 597 | . . . . . . . . . 10 |
17 | biorf 663 | . . . . . . . . . 10 | |
18 | 16, 17 | syl 14 | . . . . . . . . 9 |
19 | 5, 18 | bitrd 177 | . . . . . . . 8 |
20 | 3orrot 891 | . . . . . . . . 9 | |
21 | df-3or 886 | . . . . . . . . 9 | |
22 | 20, 21 | bitri 173 | . . . . . . . 8 |
23 | 19, 22 | syl6bbr 187 | . . . . . . 7 |
24 | 23 | eubidv 1908 | . . . . . 6 |
25 | 4, 24 | mpbii 136 | . . . . 5 |
26 | eueq3dc.3 | . . . . . . 7 | |
27 | 26 | eueq1 2713 | . . . . . 6 |
28 | ibar 285 | . . . . . . . . 9 | |
29 | 8 | adantr 261 | . . . . . . . . . . . 12 |
30 | pm2.46 658 | . . . . . . . . . . . . 13 | |
31 | 30 | adantr 261 | . . . . . . . . . . . 12 |
32 | 29, 31 | jaoi 636 | . . . . . . . . . . 11 |
33 | 32 | con2i 557 | . . . . . . . . . 10 |
34 | biorf 663 | . . . . . . . . . 10 | |
35 | 33, 34 | syl 14 | . . . . . . . . 9 |
36 | 28, 35 | bitrd 177 | . . . . . . . 8 |
37 | df-3or 886 | . . . . . . . 8 | |
38 | 36, 37 | syl6bbr 187 | . . . . . . 7 |
39 | 38 | eubidv 1908 | . . . . . 6 |
40 | 27, 39 | mpbii 136 | . . . . 5 |
41 | 25, 40 | jaoi 636 | . . . 4 |
42 | eueq3dc.2 | . . . . . 6 | |
43 | 42 | eueq1 2713 | . . . . 5 |
44 | ibar 285 | . . . . . . . 8 | |
45 | simpl 102 | . . . . . . . . . . 11 | |
46 | simpl 102 | . . . . . . . . . . 11 | |
47 | 45, 46 | orim12i 676 | . . . . . . . . . 10 |
48 | 47 | con3i 562 | . . . . . . . . 9 |
49 | biorf 663 | . . . . . . . . 9 | |
50 | 48, 49 | syl 14 | . . . . . . . 8 |
51 | 44, 50 | bitrd 177 | . . . . . . 7 |
52 | 3orcomb 894 | . . . . . . . 8 | |
53 | df-3or 886 | . . . . . . . 8 | |
54 | 52, 53 | bitri 173 | . . . . . . 7 |
55 | 51, 54 | syl6bbr 187 | . . . . . 6 |
56 | 55 | eubidv 1908 | . . . . 5 |
57 | 43, 56 | mpbii 136 | . . . 4 |
58 | 41, 57 | jaoi 636 | . . 3 |
59 | 2, 58 | sylbi 114 | . 2 DECID |
60 | 1, 59 | syl6 29 | 1 DECID DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 DECID wdc 742 w3o 884 wceq 1243 wcel 1393 weu 1900 cvv 2557 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-v 2559 |
This theorem is referenced by: moeq3dc 2717 |
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