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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 1508 | . . . . 5 | |
2 | nfv 1508 | . . . . . . 7 | |
3 | nfs1v 1912 | . . . . . . 7 | |
4 | 2, 3 | nfan 1544 | . . . . . 6 |
5 | 4 | nfex 1616 | . . . . 5 |
6 | opeq1 3705 | . . . . . . . 8 | |
7 | 6 | eqeq2d 2151 | . . . . . . 7 |
8 | sbequ12 1744 | . . . . . . 7 | |
9 | 7, 8 | anbi12d 464 | . . . . . 6 |
10 | 9 | exbidv 1797 | . . . . 5 |
11 | 1, 5, 10 | cbvex 1729 | . . . 4 |
12 | nfv 1508 | . . . . . . 7 | |
13 | cbvopab1.1 | . . . . . . . 8 | |
14 | 13 | nfsb 1919 | . . . . . . 7 |
15 | 12, 14 | nfan 1544 | . . . . . 6 |
16 | 15 | nfex 1616 | . . . . 5 |
17 | nfv 1508 | . . . . 5 | |
18 | opeq1 3705 | . . . . . . . 8 | |
19 | 18 | eqeq2d 2151 | . . . . . . 7 |
20 | sbequ 1812 | . . . . . . . 8 | |
21 | cbvopab1.2 | . . . . . . . . 9 | |
22 | cbvopab1.3 | . . . . . . . . 9 | |
23 | 21, 22 | sbie 1764 | . . . . . . . 8 |
24 | 20, 23 | syl6bb 195 | . . . . . . 7 |
25 | 19, 24 | anbi12d 464 | . . . . . 6 |
26 | 25 | exbidv 1797 | . . . . 5 |
27 | 16, 17, 26 | cbvex 1729 | . . . 4 |
28 | 11, 27 | bitri 183 | . . 3 |
29 | 28 | abbii 2255 | . 2 |
30 | df-opab 3990 | . 2 | |
31 | df-opab 3990 | . 2 | |
32 | 29, 30, 31 | 3eqtr4i 2170 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wnf 1436 wex 1468 wsb 1735 cab 2125 cop 3530 copab 3988 |
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-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-opab 3990 |
This theorem is referenced by: cbvopab1v 4004 cbvmptf 4022 cbvmpt 4023 |
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