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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 904 | . 2 DECID DECID DECID | |
2 | df-dc 805 | . . 3 DECID | |
3 | eueq3dc.1 | . . . . . . 7 | |
4 | 3 | eueq1 2829 | . . . . . 6 |
5 | ibar 299 | . . . . . . . . 9 | |
6 | pm2.45 712 | . . . . . . . . . . . . 13 | |
7 | eueq3dc.4 | . . . . . . . . . . . . . . 15 | |
8 | 7 | imnani 665 | . . . . . . . . . . . . . 14 |
9 | 8 | con2i 601 | . . . . . . . . . . . . 13 |
10 | 6, 9 | jaoi 690 | . . . . . . . . . . . 12 |
11 | 10 | con2i 601 | . . . . . . . . . . 11 |
12 | 6 | con2i 601 | . . . . . . . . . . . . 13 |
13 | 12 | bianfd 917 | . . . . . . . . . . . 12 |
14 | 8 | bianfd 917 | . . . . . . . . . . . 12 |
15 | 13, 14 | orbi12d 767 | . . . . . . . . . . 11 |
16 | 11, 15 | mtbid 646 | . . . . . . . . . 10 |
17 | biorf 718 | . . . . . . . . . 10 | |
18 | 16, 17 | syl 14 | . . . . . . . . 9 |
19 | 5, 18 | bitrd 187 | . . . . . . . 8 |
20 | 3orrot 953 | . . . . . . . . 9 | |
21 | df-3or 948 | . . . . . . . . 9 | |
22 | 20, 21 | bitri 183 | . . . . . . . 8 |
23 | 19, 22 | syl6bbr 197 | . . . . . . 7 |
24 | 23 | eubidv 1985 | . . . . . 6 |
25 | 4, 24 | mpbii 147 | . . . . 5 |
26 | eueq3dc.3 | . . . . . . 7 | |
27 | 26 | eueq1 2829 | . . . . . 6 |
28 | ibar 299 | . . . . . . . . 9 | |
29 | 8 | adantr 274 | . . . . . . . . . . . 12 |
30 | pm2.46 713 | . . . . . . . . . . . . 13 | |
31 | 30 | adantr 274 | . . . . . . . . . . . 12 |
32 | 29, 31 | jaoi 690 | . . . . . . . . . . 11 |
33 | 32 | con2i 601 | . . . . . . . . . 10 |
34 | biorf 718 | . . . . . . . . . 10 | |
35 | 33, 34 | syl 14 | . . . . . . . . 9 |
36 | 28, 35 | bitrd 187 | . . . . . . . 8 |
37 | df-3or 948 | . . . . . . . 8 | |
38 | 36, 37 | syl6bbr 197 | . . . . . . 7 |
39 | 38 | eubidv 1985 | . . . . . 6 |
40 | 27, 39 | mpbii 147 | . . . . 5 |
41 | 25, 40 | jaoi 690 | . . . 4 |
42 | eueq3dc.2 | . . . . . 6 | |
43 | 42 | eueq1 2829 | . . . . 5 |
44 | ibar 299 | . . . . . . . 8 | |
45 | simpl 108 | . . . . . . . . . . 11 | |
46 | simpl 108 | . . . . . . . . . . 11 | |
47 | 45, 46 | orim12i 733 | . . . . . . . . . 10 |
48 | 47 | con3i 606 | . . . . . . . . 9 |
49 | biorf 718 | . . . . . . . . 9 | |
50 | 48, 49 | syl 14 | . . . . . . . 8 |
51 | 44, 50 | bitrd 187 | . . . . . . 7 |
52 | 3orcomb 956 | . . . . . . . 8 | |
53 | df-3or 948 | . . . . . . . 8 | |
54 | 52, 53 | bitri 183 | . . . . . . 7 |
55 | 51, 54 | syl6bbr 197 | . . . . . 6 |
56 | 55 | eubidv 1985 | . . . . 5 |
57 | 43, 56 | mpbii 147 | . . . 4 |
58 | 41, 57 | jaoi 690 | . . 3 |
59 | 2, 58 | sylbi 120 | . 2 DECID |
60 | 1, 59 | syl6 33 | 1 DECID DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 DECID wdc 804 w3o 946 wceq 1316 wcel 1465 weu 1977 cvv 2660 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-v 2662 |
This theorem is referenced by: moeq3dc 2833 |
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