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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 1969. Other variants of this theorem are dvelimf 1968 (with no distinct variable restrictions) and dvelimALT 1963 (that avoids ax-10 1468). (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 1491 | . 2 | |
3 | dvelim.2 | . 2 | |
4 | 1, 2, 3 | dvelimf 1968 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 104 wal 1314 |
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 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 |
This theorem is referenced by: (None) |
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