Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3652 | . . 3 | |
2 | eleq2 2234 | . . . . . 6 | |
3 | abid 2158 | . . . . . 6 | |
4 | velsn 3600 | . . . . . 6 | |
5 | 2, 3, 4 | 3bitr3g 221 | . . . . 5 |
6 | nfre1 2513 | . . . . . . . . 9 | |
7 | 6 | nfab 2317 | . . . . . . . 8 |
8 | 7 | nfeq1 2322 | . . . . . . 7 |
9 | eusvobj1.1 | . . . . . . . . 9 | |
10 | 9 | elabrex 5737 | . . . . . . . 8 |
11 | eleq2 2234 | . . . . . . . . 9 | |
12 | 9 | elsn 3599 | . . . . . . . . . 10 |
13 | eqcom 2172 | . . . . . . . . . 10 | |
14 | 12, 13 | bitri 183 | . . . . . . . . 9 |
15 | 11, 14 | bitrdi 195 | . . . . . . . 8 |
16 | 10, 15 | syl5ib 153 | . . . . . . 7 |
17 | 8, 16 | ralrimi 2541 | . . . . . 6 |
18 | eqeq1 2177 | . . . . . . 7 | |
19 | 18 | ralbidv 2470 | . . . . . 6 |
20 | 17, 19 | syl5ibrcom 156 | . . . . 5 |
21 | 5, 20 | sylbid 149 | . . . 4 |
22 | 21 | exlimiv 1591 | . . 3 |
23 | 1, 22 | sylbi 120 | . 2 |
24 | euex 2049 | . . 3 | |
25 | rexm 3514 | . . . 4 | |
26 | 25 | exlimiv 1591 | . . 3 |
27 | r19.2m 3501 | . . . 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 1348 wex 1485 weu 2019 wcel 2141 cab 2156 wral 2448 wrex 2449 cvv 2730 csn 3583 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-csb 3050 df-sn 3589 |
This theorem is referenced by: eusvobj1 5840 |
Copyright terms: Public domain | W3C validator |