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Mirrors > Home > ILE Home > Th. List > eueq2dc | Unicode version |
Description: Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.) |
Ref | Expression |
---|---|
eueq2dc.1 |
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eueq2dc.2 |
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Ref | Expression |
---|---|
eueq2dc |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 821 |
. 2
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2 | notnot 619 |
. . . . 5
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3 | eueq2dc.1 |
. . . . . . 7
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4 | 3 | eueq1 2860 |
. . . . . 6
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5 | euanv 2057 |
. . . . . . 7
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6 | 5 | biimpri 132 |
. . . . . 6
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7 | 4, 6 | mpan2 422 |
. . . . 5
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8 | euorv 2027 |
. . . . 5
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9 | 2, 7, 8 | syl2anc 409 |
. . . 4
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10 | orcom 718 |
. . . . . 6
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11 | 2 | bianfd 933 |
. . . . . . 7
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12 | 11 | orbi2d 780 |
. . . . . 6
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13 | 10, 12 | syl5bb 191 |
. . . . 5
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14 | 13 | eubidv 2008 |
. . . 4
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15 | 9, 14 | mpbid 146 |
. . 3
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16 | eueq2dc.2 |
. . . . . . 7
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17 | 16 | eueq1 2860 |
. . . . . 6
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18 | euanv 2057 |
. . . . . . 7
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19 | 18 | biimpri 132 |
. . . . . 6
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20 | 17, 19 | mpan2 422 |
. . . . 5
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21 | euorv 2027 |
. . . . 5
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22 | 20, 21 | mpdan 418 |
. . . 4
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23 | id 19 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 23 | bianfd 933 |
. . . . . 6
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25 | 24 | orbi1d 781 |
. . . . 5
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26 | 25 | eubidv 2008 |
. . . 4
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27 | 22, 26 | mpbid 146 |
. . 3
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28 | 15, 27 | jaoi 706 |
. 2
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29 | 1, 28 | sylbi 120 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-v 2691 |
This theorem is referenced by: (None) |
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