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Mirrors > Home > ILE Home > Th. List > cbvopab1 | Unicode version |
Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
cbvopab1.1 | |
cbvopab1.2 | |
cbvopab1.3 |
Ref | Expression |
---|---|
cbvopab1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1421 | . . . . 5 | |
2 | nfv 1421 | . . . . . . 7 | |
3 | nfs1v 1815 | . . . . . . 7 | |
4 | 2, 3 | nfan 1457 | . . . . . 6 |
5 | 4 | nfex 1528 | . . . . 5 |
6 | opeq1 3549 | . . . . . . . 8 | |
7 | 6 | eqeq2d 2051 | . . . . . . 7 |
8 | sbequ12 1654 | . . . . . . 7 | |
9 | 7, 8 | anbi12d 442 | . . . . . 6 |
10 | 9 | exbidv 1706 | . . . . 5 |
11 | 1, 5, 10 | cbvex 1639 | . . . 4 |
12 | nfv 1421 | . . . . . . 7 | |
13 | cbvopab1.1 | . . . . . . . 8 | |
14 | 13 | nfsb 1822 | . . . . . . 7 |
15 | 12, 14 | nfan 1457 | . . . . . 6 |
16 | 15 | nfex 1528 | . . . . 5 |
17 | nfv 1421 | . . . . 5 | |
18 | opeq1 3549 | . . . . . . . 8 | |
19 | 18 | eqeq2d 2051 | . . . . . . 7 |
20 | sbequ 1721 | . . . . . . . 8 | |
21 | cbvopab1.2 | . . . . . . . . 9 | |
22 | cbvopab1.3 | . . . . . . . . 9 | |
23 | 21, 22 | sbie 1674 | . . . . . . . 8 |
24 | 20, 23 | syl6bb 185 | . . . . . . 7 |
25 | 19, 24 | anbi12d 442 | . . . . . 6 |
26 | 25 | exbidv 1706 | . . . . 5 |
27 | 16, 17, 26 | cbvex 1639 | . . . 4 |
28 | 11, 27 | bitri 173 | . . 3 |
29 | 28 | abbii 2153 | . 2 |
30 | df-opab 3819 | . 2 | |
31 | df-opab 3819 | . 2 | |
32 | 29, 30, 31 | 3eqtr4i 2070 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wnf 1349 wex 1381 wsb 1645 cab 2026 cop 3378 copab 3817 |
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-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819 |
This theorem is referenced by: cbvopab1v 3833 cbvmpt 3851 |
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