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Mirrors > Home > ILE Home > Th. List > eueq2dc | Unicode version |
Description: Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.) |
Ref | Expression |
---|---|
eueq2dc.1 | |
eueq2dc.2 |
Ref | Expression |
---|---|
eueq2dc | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 743 | . 2 DECID | |
2 | notnot 559 | . . . . 5 | |
3 | eueq2dc.1 | . . . . . . 7 | |
4 | 3 | eueq1 2713 | . . . . . 6 |
5 | euanv 1957 | . . . . . . 7 | |
6 | 5 | biimpri 124 | . . . . . 6 |
7 | 4, 6 | mpan2 401 | . . . . 5 |
8 | euorv 1927 | . . . . 5 | |
9 | 2, 7, 8 | syl2anc 391 | . . . 4 |
10 | orcom 647 | . . . . . 6 | |
11 | 2 | bianfd 855 | . . . . . . 7 |
12 | 11 | orbi2d 704 | . . . . . 6 |
13 | 10, 12 | syl5bb 181 | . . . . 5 |
14 | 13 | eubidv 1908 | . . . 4 |
15 | 9, 14 | mpbid 135 | . . 3 |
16 | eueq2dc.2 | . . . . . . 7 | |
17 | 16 | eueq1 2713 | . . . . . 6 |
18 | euanv 1957 | . . . . . . 7 | |
19 | 18 | biimpri 124 | . . . . . 6 |
20 | 17, 19 | mpan2 401 | . . . . 5 |
21 | euorv 1927 | . . . . 5 | |
22 | 20, 21 | mpdan 398 | . . . 4 |
23 | id 19 | . . . . . . 7 | |
24 | 23 | bianfd 855 | . . . . . 6 |
25 | 24 | orbi1d 705 | . . . . 5 |
26 | 25 | eubidv 1908 | . . . 4 |
27 | 22, 26 | mpbid 135 | . . 3 |
28 | 15, 27 | jaoi 636 | . 2 |
29 | 1, 28 | sylbi 114 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wo 629 DECID wdc 742 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-tru 1246 df-fal 1249 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: (None) |
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