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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 1489 | . . . . . 6 | |
2 | orcom 718 | . . . . . . 7 | |
3 | 2 | orbi2i 752 | . . . . . 6 |
4 | 1, 3 | mpbi 144 | . . . . 5 |
5 | orass 757 | . . . . 5 | |
6 | 4, 5 | mpbir 145 | . . . 4 |
7 | nfae 1699 | . . . . . . 7 | |
8 | a16nf 1846 | . . . . . . 7 | |
9 | 7, 8 | alrimi 1502 | . . . . . 6 |
10 | df-nf 1441 | . . . . . . . 8 | |
11 | id 19 | . . . . . . . . 9 | |
12 | dvelimor.1 | . . . . . . . . . 10 | |
13 | 12 | a1i 9 | . . . . . . . . 9 |
14 | 11, 13 | nfimd 1565 | . . . . . . . 8 |
15 | 10, 14 | sylbir 134 | . . . . . . 7 |
16 | 15 | alimi 1435 | . . . . . 6 |
17 | 9, 16 | jaoi 706 | . . . . 5 |
18 | 17 | orim1i 750 | . . . 4 |
19 | 6, 18 | ax-mp 5 | . . 3 |
20 | orcom 718 | . . 3 | |
21 | 19, 20 | mpbi 144 | . 2 |
22 | nfalt 1558 | . . . 4 | |
23 | ax-17 1506 | . . . . . 6 | |
24 | dvelimor.2 | . . . . . 6 | |
25 | 23, 24 | equsalh 1706 | . . . . 5 |
26 | 25 | nfbii 1453 | . . . 4 |
27 | 22, 26 | sylib 121 | . . 3 |
28 | 27 | orim2i 751 | . 2 |
29 | 21, 28 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wo 698 wal 1333 wnf 1440 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-sb 1743 |
This theorem is referenced by: nfsb4or 2001 rgen2a 2511 |
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