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Theorem dvelimor 1891
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.)
Hypotheses
Ref Expression
dvelimor.1  F/
dvelimor.2
Assertion
Ref Expression
dvelimor  F/
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,,)   (,)

Proof of Theorem dvelimor
StepHypRef Expression
1 ax-bnd 1396 . . . . . 6
2 orcom 646 . . . . . . 7
32orbi2i 678 . . . . . 6
41, 3mpbi 133 . . . . 5
5 orass 683 . . . . 5
64, 5mpbir 134 . . . 4
7 nfae 1604 . . . . . . 7  F/
8 a16nf 1743 . . . . . . 7  F/
97, 8alrimi 1412 . . . . . 6  F/
10 df-nf 1347 . . . . . . . 8  F/
11 id 19 . . . . . . . . 9  F/  F/
12 dvelimor.1 . . . . . . . . . 10  F/
1312a1i 9 . . . . . . . . 9  F/  F/
1411, 13nfimd 1474 . . . . . . . 8  F/  F/
1510, 14sylbir 125 . . . . . . 7  F/
1615alimi 1341 . . . . . 6  F/
179, 16jaoi 635 . . . . 5  F/
1817orim1i 676 . . . 4  F/
196, 18ax-mp 7 . . 3  F/
20 orcom 646 . . 3  F/  F/
2119, 20mpbi 133 . 2  F/
22 nfalt 1467 . . . 4  F/  F/
23 ax-17 1416 . . . . . 6
24 dvelimor.2 . . . . . 6
2523, 24equsalh 1611 . . . . 5
2625nfbii 1359 . . . 4  F/  F/
2722, 26sylib 127 . . 3  F/  F/
2827orim2i 677 . 2  F/  F/
2921, 28ax-mp 7 1  F/
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   wo 628  wal 1240   F/wnf 1346
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by:  nfsb4or  1896  rgen2a  2369
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