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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 1971). (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 1493 | . . . . 5 | |
2 | eleq1 2180 | . . . . 5 | |
3 | 1, 2 | dvelimor 1971 | . . . 4 |
4 | eleq1 2180 | . . . . . . . . 9 | |
5 | rgen2a.1 | . . . . . . . . . 10 | |
6 | 5 | ex 114 | . . . . . . . . 9 |
7 | 4, 6 | syl6bi 162 | . . . . . . . 8 |
8 | 7 | pm2.43d 50 | . . . . . . 7 |
9 | 8 | alimi 1416 | . . . . . 6 |
10 | 9 | a1d 22 | . . . . 5 |
11 | nfr 1483 | . . . . . 6 | |
12 | 6 | alimi 1416 | . . . . . 6 |
13 | 11, 12 | syl6 33 | . . . . 5 |
14 | 10, 13 | jaoi 690 | . . . 4 |
15 | 3, 14 | ax-mp 5 | . . 3 |
16 | df-ral 2398 | . . 3 | |
17 | 15, 16 | sylibr 133 | . 2 |
18 | 17 | rgen 2462 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 682 wal 1314 wceq 1316 wnf 1421 wcel 1465 wral 2393 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-cleq 2110 df-clel 2113 df-ral 2398 |
This theorem is referenced by: ordsucunielexmid 4416 onintexmid 4457 isoid 5679 issmo 6153 oawordriexmid 6334 ecopover 6495 ecopoverg 6498 1domsn 6681 unfiexmid 6774 axaddf 7644 axmulf 7645 subf 7932 negiso 8681 cnref1o 9408 xaddf 9595 ioof 9722 fzof 9889 xrnegiso 10999 reeff1 11334 gcdf 11588 eucalgf 11663 qredeu 11705 qnnen 11871 strsetsid 11919 hmeofn 12398 ismeti 12442 qtopbasss 12617 tgqioo 12643 peano4nninf 13127 |
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