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Mirrors > Home > ILE Home > Th. List > mhmlem | Unicode version |
Description: Lemma for mhmmnd 12869 and ghmgrp 12871. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 25-Jan-2020.) |
Ref | Expression |
---|---|
ghmgrp.f |
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mhmlem.a |
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mhmlem.b |
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Ref | Expression |
---|---|
mhmlem |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 |
. 2
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2 | mhmlem.a |
. 2
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3 | mhmlem.b |
. 2
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4 | eleq1 2240 |
. . . . . 6
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5 | 4 | 3anbi2d 1317 |
. . . . 5
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6 | fvoveq1 5892 |
. . . . . 6
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7 | fveq2 5511 |
. . . . . . 7
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8 | 7 | oveq1d 5884 |
. . . . . 6
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9 | 6, 8 | eqeq12d 2192 |
. . . . 5
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10 | 5, 9 | imbi12d 234 |
. . . 4
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11 | eleq1 2240 |
. . . . . 6
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12 | 11 | 3anbi3d 1318 |
. . . . 5
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13 | oveq2 5877 |
. . . . . . 7
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14 | 13 | fveq2d 5515 |
. . . . . 6
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15 | fveq2 5511 |
. . . . . . 7
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16 | 15 | oveq2d 5885 |
. . . . . 6
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17 | 14, 16 | eqeq12d 2192 |
. . . . 5
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18 | 12, 17 | imbi12d 234 |
. . . 4
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19 | ghmgrp.f |
. . . 4
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20 | 10, 18, 19 | vtocl2g 2801 |
. . 3
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21 | 2, 3, 20 | syl2anc 411 |
. 2
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22 | 1, 2, 3, 21 | mp3and 1340 |
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 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2739 df-un 3133 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-iota 5174 df-fv 5220 df-ov 5872 |
This theorem is referenced by: mhmid 12868 mhmmnd 12869 ghmgrp 12871 |
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