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Mirrors > Home > ILE Home > Th. List > ereq1 | Unicode version |
Description: Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ereq1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | releq 4629 |
. . 3
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2 | dmeq 4747 |
. . . 4
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3 | 2 | eqeq1d 2149 |
. . 3
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4 | cnveq 4721 |
. . . . . 6
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5 | coeq1 4704 |
. . . . . . 7
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6 | coeq2 4705 |
. . . . . . 7
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7 | 5, 6 | eqtrd 2173 |
. . . . . 6
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8 | 4, 7 | uneq12d 3236 |
. . . . 5
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9 | 8 | sseq1d 3131 |
. . . 4
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10 | sseq2 3126 |
. . . 4
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11 | 9, 10 | bitrd 187 |
. . 3
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12 | 1, 3, 11 | 3anbi123d 1291 |
. 2
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13 | df-er 6437 |
. 2
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14 | df-er 6437 |
. 2
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15 | 12, 13, 14 | 3bitr4g 222 |
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-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-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-er 6437 |
This theorem is referenced by: riinerm 6510 |
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