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Mirrors > Home > ILE Home > Th. List > elabgt | Unicode version |
Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2775.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Ref | Expression |
---|---|
elabgt |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abid 2083 |
. . . . . . 7
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2 | eleq1 2157 |
. . . . . . 7
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3 | 1, 2 | syl5bbr 193 |
. . . . . 6
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4 | 3 | bibi1d 232 |
. . . . 5
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5 | 4 | biimpd 143 |
. . . 4
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6 | 5 | a2i 11 |
. . 3
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7 | 6 | alimi 1396 |
. 2
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8 | nfcv 2235 |
. . . 4
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9 | nfab1 2237 |
. . . . . 6
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10 | 9 | nfel2 2248 |
. . . . 5
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11 | nfv 1473 |
. . . . 5
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12 | 10, 11 | nfbi 1533 |
. . . 4
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13 | pm5.5 241 |
. . . 4
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14 | 8, 12, 13 | spcgf 2715 |
. . 3
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15 | 14 | imp 123 |
. 2
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16 | 7, 15 | sylan2 281 |
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 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 |
This theorem is referenced by: elrab3t 2784 |
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