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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 820 | . 2 DECID | |
2 | notnot 618 | . . . . 5 | |
3 | eueq2dc.1 | . . . . . . 7 | |
4 | 3 | eueq1 2851 | . . . . . 6 |
5 | euanv 2054 | . . . . . . 7 | |
6 | 5 | biimpri 132 | . . . . . 6 |
7 | 4, 6 | mpan2 421 | . . . . 5 |
8 | euorv 2024 | . . . . 5 | |
9 | 2, 7, 8 | syl2anc 408 | . . . 4 |
10 | orcom 717 | . . . . . 6 | |
11 | 2 | bianfd 932 | . . . . . . 7 |
12 | 11 | orbi2d 779 | . . . . . 6 |
13 | 10, 12 | syl5bb 191 | . . . . 5 |
14 | 13 | eubidv 2005 | . . . 4 |
15 | 9, 14 | mpbid 146 | . . 3 |
16 | eueq2dc.2 | . . . . . . 7 | |
17 | 16 | eueq1 2851 | . . . . . 6 |
18 | euanv 2054 | . . . . . . 7 | |
19 | 18 | biimpri 132 | . . . . . 6 |
20 | 17, 19 | mpan2 421 | . . . . 5 |
21 | euorv 2024 | . . . . 5 | |
22 | 20, 21 | mpdan 417 | . . . 4 |
23 | id 19 | . . . . . . 7 | |
24 | 23 | bianfd 932 | . . . . . 6 |
25 | 24 | orbi1d 780 | . . . . 5 |
26 | 25 | eubidv 2005 | . . . 4 |
27 | 22, 26 | mpbid 146 | . . 3 |
28 | 15, 27 | jaoi 705 | . 2 |
29 | 1, 28 | sylbi 120 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 DECID wdc 819 wceq 1331 wcel 1480 weu 1997 cvv 2681 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-v 2683 |
This theorem is referenced by: (None) |
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