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Mirrors > Home > ILE Home > Th. List > dvelimor | Unicode version |
Description: Disjunctive distinct variable constraint elimination. A user of this theorem starts with a formula (containing ) and a distinct variable constraint between and . The theorem makes it possible to replace the distinct variable constraint with the disjunct ( is just a version of with substituted for ). (Contributed by Jim Kingdon, 11-May-2018.) |
Ref | Expression |
---|---|
dvelimor.1 | |
dvelimor.2 |
Ref | Expression |
---|---|
dvelimor |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-bndl 1399 | . . . . . 6 | |
2 | orcom 647 | . . . . . . 7 | |
3 | 2 | orbi2i 679 | . . . . . 6 |
4 | 1, 3 | mpbi 133 | . . . . 5 |
5 | orass 684 | . . . . 5 | |
6 | 4, 5 | mpbir 134 | . . . 4 |
7 | nfae 1607 | . . . . . . 7 | |
8 | a16nf 1746 | . . . . . . 7 | |
9 | 7, 8 | alrimi 1415 | . . . . . 6 |
10 | df-nf 1350 | . . . . . . . 8 | |
11 | id 19 | . . . . . . . . 9 | |
12 | dvelimor.1 | . . . . . . . . . 10 | |
13 | 12 | a1i 9 | . . . . . . . . 9 |
14 | 11, 13 | nfimd 1477 | . . . . . . . 8 |
15 | 10, 14 | sylbir 125 | . . . . . . 7 |
16 | 15 | alimi 1344 | . . . . . 6 |
17 | 9, 16 | jaoi 636 | . . . . 5 |
18 | 17 | orim1i 677 | . . . 4 |
19 | 6, 18 | ax-mp 7 | . . 3 |
20 | orcom 647 | . . 3 | |
21 | 19, 20 | mpbi 133 | . 2 |
22 | nfalt 1470 | . . . 4 | |
23 | ax-17 1419 | . . . . . 6 | |
24 | dvelimor.2 | . . . . . 6 | |
25 | 23, 24 | equsalh 1614 | . . . . 5 |
26 | 25 | nfbii 1362 | . . . 4 |
27 | 22, 26 | sylib 127 | . . 3 |
28 | 27 | orim2i 678 | . 2 |
29 | 21, 28 | ax-mp 7 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 98 wo 629 wal 1241 wnf 1349 |
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 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 |
This theorem is referenced by: nfsb4or 1899 rgen2a 2375 |
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