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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 3592 | . . 3 | |
2 | eleq2 2203 | . . . . . 6 | |
3 | abid 2127 | . . . . . 6 | |
4 | velsn 3544 | . . . . . 6 | |
5 | 2, 3, 4 | 3bitr3g 221 | . . . . 5 |
6 | nfre1 2476 | . . . . . . . . 9 | |
7 | 6 | nfab 2286 | . . . . . . . 8 |
8 | 7 | nfeq1 2291 | . . . . . . 7 |
9 | eusvobj1.1 | . . . . . . . . 9 | |
10 | 9 | elabrex 5659 | . . . . . . . 8 |
11 | eleq2 2203 | . . . . . . . . 9 | |
12 | 9 | elsn 3543 | . . . . . . . . . 10 |
13 | eqcom 2141 | . . . . . . . . . 10 | |
14 | 12, 13 | bitri 183 | . . . . . . . . 9 |
15 | 11, 14 | syl6bb 195 | . . . . . . . 8 |
16 | 10, 15 | syl5ib 153 | . . . . . . 7 |
17 | 8, 16 | ralrimi 2503 | . . . . . 6 |
18 | eqeq1 2146 | . . . . . . 7 | |
19 | 18 | ralbidv 2437 | . . . . . 6 |
20 | 17, 19 | syl5ibrcom 156 | . . . . 5 |
21 | 5, 20 | sylbid 149 | . . . 4 |
22 | 21 | exlimiv 1577 | . . 3 |
23 | 1, 22 | sylbi 120 | . 2 |
24 | euex 2029 | . . 3 | |
25 | rexm 3462 | . . . 4 | |
26 | 25 | exlimiv 1577 | . . 3 |
27 | r19.2m 3449 | . . . 4 | |
28 | 27 | ex 114 | . . 3 |
29 | 24, 26, 28 | 3syl 17 | . 2 |
30 | 23, 29 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1331 wex 1468 wcel 1480 weu 1999 cab 2125 wral 2416 wrex 2417 cvv 2686 csn 3527 |
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-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 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-csb 3004 df-sn 3533 |
This theorem is referenced by: eusvobj1 5761 |
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