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Mirrors > Home > ILE Home > Th. List > cbvopab1s | Unicode version |
Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.) |
Ref | Expression |
---|---|
cbvopab1s |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1515 | . . . 4 | |
2 | nfv 1515 | . . . . . 6 | |
3 | nfs1v 1926 | . . . . . 6 | |
4 | 2, 3 | nfan 1552 | . . . . 5 |
5 | 4 | nfex 1624 | . . . 4 |
6 | opeq1 3752 | . . . . . . 7 | |
7 | 6 | eqeq2d 2176 | . . . . . 6 |
8 | sbequ12 1758 | . . . . . 6 | |
9 | 7, 8 | anbi12d 465 | . . . . 5 |
10 | 9 | exbidv 1812 | . . . 4 |
11 | 1, 5, 10 | cbvex 1743 | . . 3 |
12 | 11 | abbii 2280 | . 2 |
13 | df-opab 4038 | . 2 | |
14 | df-opab 4038 | . 2 | |
15 | 12, 13, 14 | 3eqtr4i 2195 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1342 wex 1479 wsb 1749 cab 2150 cop 3573 copab 4036 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-sn 3576 df-pr 3577 df-op 3579 df-opab 4038 |
This theorem is referenced by: (None) |
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