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Mirrors > Home > ILE Home > Th. List > rgen2a | Unicode version |
Description: Generalization rule for restricted quantification. Note that and needn't be distinct (and illustrates the use of dvelimor 1894). (Contributed by NM, 23-Nov-1994.) (Proof rewritten by Jim Kingdon, 1-Jun-2018.) |
Ref | Expression |
---|---|
rgen2a.1 |
Ref | Expression |
---|---|
rgen2a |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1421 | . . . . 5 | |
2 | eleq1 2100 | . . . . 5 | |
3 | 1, 2 | dvelimor 1894 | . . . 4 |
4 | eleq1 2100 | . . . . . . . . 9 | |
5 | rgen2a.1 | . . . . . . . . . 10 | |
6 | 5 | ex 108 | . . . . . . . . 9 |
7 | 4, 6 | syl6bi 152 | . . . . . . . 8 |
8 | 7 | pm2.43d 44 | . . . . . . 7 |
9 | 8 | alimi 1344 | . . . . . 6 |
10 | 9 | a1d 22 | . . . . 5 |
11 | nfr 1411 | . . . . . 6 | |
12 | 6 | alimi 1344 | . . . . . 6 |
13 | 11, 12 | syl6 29 | . . . . 5 |
14 | 10, 13 | jaoi 636 | . . . 4 |
15 | 3, 14 | ax-mp 7 | . . 3 |
16 | df-ral 2311 | . . 3 | |
17 | 15, 16 | sylibr 137 | . 2 |
18 | 17 | rgen 2374 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wo 629 wal 1241 wceq 1243 wnf 1349 wcel 1393 wral 2306 |
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-nf 1350 df-sb 1646 df-cleq 2033 df-clel 2036 df-ral 2311 |
This theorem is referenced by: ordsucunielexmid 4256 onintexmid 4297 isoid 5450 issmo 5903 ecopover 6204 ecopoverg 6207 subf 7213 cnref1o 8582 ioof 8840 fzof 9001 |
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