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Mirrors > Home > ILE Home > Th. List > eusvobj2 | Unicode version |
Description: Specify the same property in two ways when class is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
eusvobj1.1 |
Ref | Expression |
---|---|
eusvobj2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euabsn2 3439 | . . 3 | |
2 | eleq2 2101 | . . . . . 6 | |
3 | abid 2028 | . . . . . 6 | |
4 | velsn 3392 | . . . . . 6 | |
5 | 2, 3, 4 | 3bitr3g 211 | . . . . 5 |
6 | nfre1 2365 | . . . . . . . . 9 | |
7 | 6 | nfab 2182 | . . . . . . . 8 |
8 | 7 | nfeq1 2187 | . . . . . . 7 |
9 | eusvobj1.1 | . . . . . . . . 9 | |
10 | 9 | elabrex 5397 | . . . . . . . 8 |
11 | eleq2 2101 | . . . . . . . . 9 | |
12 | 9 | elsn 3391 | . . . . . . . . . 10 |
13 | eqcom 2042 | . . . . . . . . . 10 | |
14 | 12, 13 | bitri 173 | . . . . . . . . 9 |
15 | 11, 14 | syl6bb 185 | . . . . . . . 8 |
16 | 10, 15 | syl5ib 143 | . . . . . . 7 |
17 | 8, 16 | ralrimi 2390 | . . . . . 6 |
18 | eqeq1 2046 | . . . . . . 7 | |
19 | 18 | ralbidv 2326 | . . . . . 6 |
20 | 17, 19 | syl5ibrcom 146 | . . . . 5 |
21 | 5, 20 | sylbid 139 | . . . 4 |
22 | 21 | exlimiv 1489 | . . 3 |
23 | 1, 22 | sylbi 114 | . 2 |
24 | euex 1930 | . . 3 | |
25 | rexm 3320 | . . . 4 | |
26 | 25 | exlimiv 1489 | . . 3 |
27 | r19.2m 3309 | . . . 4 | |
28 | 27 | ex 108 | . . 3 |
29 | 24, 26, 28 | 3syl 17 | . 2 |
30 | 23, 29 | impbid 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 98 wceq 1243 wex 1381 wcel 1393 weu 1900 cab 2026 wral 2306 wrex 2307 cvv 2557 csn 3375 |
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-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-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-csb 2853 df-sn 3381 |
This theorem is referenced by: eusvobj1 5499 |
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