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Mirrors > Home > ILE Home > Th. List > cleqh | Unicode version |
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2306. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
cleqh.1 |
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cleqh.2 |
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Ref | Expression |
---|---|
cleqh |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2134 |
. 2
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2 | ax-17 1507 |
. . . 4
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3 | dfbi2 386 |
. . . . 5
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4 | cleqh.1 |
. . . . . . 7
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5 | cleqh.2 |
. . . . . . 7
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6 | 4, 5 | hbim 1525 |
. . . . . 6
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7 | 5, 4 | hbim 1525 |
. . . . . 6
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8 | 6, 7 | hban 1527 |
. . . . 5
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9 | 3, 8 | hbxfrbi 1449 |
. . . 4
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10 | eleq1 2203 |
. . . . . 6
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11 | eleq1 2203 |
. . . . . 6
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12 | 10, 11 | bibi12d 234 |
. . . . 5
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13 | 12 | biimpd 143 |
. . . 4
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14 | 2, 9, 13 | cbv3h 1722 |
. . 3
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15 | 12 | equcoms 1685 |
. . . . 5
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16 | 15 | biimprd 157 |
. . . 4
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17 | 9, 2, 16 | cbv3h 1722 |
. . 3
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18 | 14, 17 | impbii 125 |
. 2
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19 | 1, 18 | bitr4i 186 |
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-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 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-nf 1438 df-cleq 2133 df-clel 2136 |
This theorem is referenced by: abeq2 2249 |
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