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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 1461 | . . . . 5 | |
2 | nfv 1461 | . . . . . . 7 | |
3 | nfs1v 1856 | . . . . . . 7 | |
4 | 2, 3 | nfan 1497 | . . . . . 6 |
5 | 4 | nfex 1568 | . . . . 5 |
6 | opeq1 3570 | . . . . . . . 8 | |
7 | 6 | eqeq2d 2092 | . . . . . . 7 |
8 | sbequ12 1694 | . . . . . . 7 | |
9 | 7, 8 | anbi12d 456 | . . . . . 6 |
10 | 9 | exbidv 1746 | . . . . 5 |
11 | 1, 5, 10 | cbvex 1679 | . . . 4 |
12 | nfv 1461 | . . . . . . 7 | |
13 | cbvopab1.1 | . . . . . . . 8 | |
14 | 13 | nfsb 1863 | . . . . . . 7 |
15 | 12, 14 | nfan 1497 | . . . . . 6 |
16 | 15 | nfex 1568 | . . . . 5 |
17 | nfv 1461 | . . . . 5 | |
18 | opeq1 3570 | . . . . . . . 8 | |
19 | 18 | eqeq2d 2092 | . . . . . . 7 |
20 | sbequ 1761 | . . . . . . . 8 | |
21 | cbvopab1.2 | . . . . . . . . 9 | |
22 | cbvopab1.3 | . . . . . . . . 9 | |
23 | 21, 22 | sbie 1714 | . . . . . . . 8 |
24 | 20, 23 | syl6bb 194 | . . . . . . 7 |
25 | 19, 24 | anbi12d 456 | . . . . . 6 |
26 | 25 | exbidv 1746 | . . . . 5 |
27 | 16, 17, 26 | cbvex 1679 | . . . 4 |
28 | 11, 27 | bitri 182 | . . 3 |
29 | 28 | abbii 2194 | . 2 |
30 | df-opab 3840 | . 2 | |
31 | df-opab 3840 | . 2 | |
32 | 29, 30, 31 | 3eqtr4i 2111 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wnf 1389 wex 1421 wsb 1685 cab 2067 cop 3401 copab 3838 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-opab 3840 |
This theorem is referenced by: cbvopab1v 3854 cbvmpt 3872 |
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