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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdsepnft | Unicode version |
Description: Closed form of bdsepnf 10008. Version of ax-bdsep 10004 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 10005 when sufficient. (Contributed by BJ, 19-Oct-2019.) |
Ref | Expression |
---|---|
bdsepnft.1 |
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Ref | Expression |
---|---|
bdsepnft |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdsepnft.1 |
. . 3
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2 | 1 | bdsep2 10006 |
. 2
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3 | nfnf1 1436 |
. . . 4
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4 | 3 | nfal 1468 |
. . 3
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5 | nfa1 1434 |
. . . 4
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6 | nfvd 1422 |
. . . . 5
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7 | nfv 1421 |
. . . . . . 7
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8 | 7 | a1i 9 |
. . . . . 6
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9 | sp 1401 |
. . . . . 6
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10 | 8, 9 | nfand 1460 |
. . . . 5
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11 | 6, 10 | nfbid 1480 |
. . . 4
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12 | 5, 11 | nfald 1643 |
. . 3
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13 | nfv 1421 |
. . . . . 6
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14 | 5, 13 | nfan 1457 |
. . . . 5
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15 | elequ2 1601 |
. . . . . . 7
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16 | 15 | adantl 262 |
. . . . . 6
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17 | 16 | bibi1d 222 |
. . . . 5
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18 | 14, 17 | albid 1506 |
. . . 4
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19 | 18 | ex 108 |
. . 3
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20 | 4, 12, 19 | cbvexd 1802 |
. 2
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21 | 2, 20 | mpbii 136 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
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-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-bdsep 10004 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-cleq 2033 df-clel 2036 |
This theorem is referenced by: bdsepnf 10008 |
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