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Mirrors > Home > ILE Home > Th. List > dvelim | Unicode version |
Description: This theorem can be used
to eliminate a distinct variable restriction on
and and replace it with the
"distinctor"
as an antecedent. normally has free and can be read
, and
substitutes for and can be read
. We don't require that and be
distinct: if
they aren't, the distinctor will become false (in multiple-element
domains of discourse) and "protect" the consequent.
To obtain a closed-theorem form of this inference, prefix the hypotheses with , conjoin them, and apply dvelimdf 1892. Other variants of this theorem are dvelimf 1891 (with no distinct variable restrictions) and dvelimALT 1886 (that avoids ax-10 1396). (Contributed by NM, 23-Nov-1994.) |
Ref | Expression |
---|---|
dvelim.1 | |
dvelim.2 |
Ref | Expression |
---|---|
dvelim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvelim.1 | . 2 | |
2 | ax-17 1419 | . 2 | |
3 | dvelim.2 | . 2 | |
4 | 1, 2, 3 | dvelimf 1891 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 98 wal 1241 |
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-in2 545 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: (None) |
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