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Mirrors > Home > ILE Home > Th. List > eqer | Unicode version |
Description: Equivalence relation
involving equality of dependent classes ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
eqer.1 |
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eqer.2 |
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Ref | Expression |
---|---|
eqer |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqer.2 |
. . . . 5
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2 | 1 | relopabi 4603 |
. . . 4
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3 | 2 | a1i 9 |
. . 3
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4 | id 19 |
. . . . . 6
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5 | 4 | eqcomd 2105 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | eqer.1 |
. . . . . 6
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7 | 6, 1 | eqerlem 6390 |
. . . . 5
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8 | 6, 1 | eqerlem 6390 |
. . . . 5
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9 | 5, 7, 8 | 3imtr4i 200 |
. . . 4
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10 | 9 | adantl 273 |
. . 3
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11 | eqtr 2117 |
. . . . 5
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12 | 6, 1 | eqerlem 6390 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | 7, 12 | anbi12i 451 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 6, 1 | eqerlem 6390 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 11, 13, 14 | 3imtr4i 200 |
. . . 4
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16 | 15 | adantl 273 |
. . 3
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17 | vex 2644 |
. . . . 5
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18 | eqid 2100 |
. . . . . 6
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19 | 6, 1 | eqerlem 6390 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 18, 19 | mpbir 145 |
. . . . 5
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21 | 17, 20 | 2th 173 |
. . . 4
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22 | 21 | a1i 9 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 3, 10, 16, 22 | iserd 6385 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 23 | mptru 1308 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-sbc 2863 df-csb 2956 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-br 3876 df-opab 3930 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-er 6359 |
This theorem is referenced by: ider 6392 |
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