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Mirrors > Home > ILE Home > Th. List > resieq | Unicode version |
Description: A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.) |
Ref | Expression |
---|---|
resieq |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 4022 |
. . . . 5
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2 | eqeq2 2199 |
. . . . 5
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3 | 1, 2 | bibi12d 235 |
. . . 4
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4 | 3 | imbi2d 230 |
. . 3
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5 | vex 2755 |
. . . . 5
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6 | 5 | opres 4931 |
. . . 4
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7 | df-br 4019 |
. . . 4
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8 | 5 | ideq 4794 |
. . . . 5
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9 | df-br 4019 |
. . . . 5
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10 | 8, 9 | bitr3i 186 |
. . . 4
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11 | 6, 7, 10 | 3bitr4g 223 |
. . 3
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12 | 4, 11 | vtoclg 2812 |
. 2
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13 | 12 | impcom 125 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-res 4653 |
This theorem is referenced by: foeqcnvco 5808 f1eqcocnv 5809 |
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