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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 1996. Other variants of this theorem are dvelimf 1995 (with no distinct variable restrictions) and dvelimALT 1990 (that avoids ax-10 1485). (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 1506 | . 2 | |
3 | dvelim.2 | . 2 | |
4 | 1, 2, 3 | dvelimf 1995 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 104 wal 1333 |
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-in2 605 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: (None) |
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