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Mirrors > Home > ILE Home > Th. List > freceq1 | Unicode version |
Description: Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
Ref | Expression |
---|---|
freceq1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 109 |
. . . . . . . . . . 11
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2 | 1 | fveq1d 5529 |
. . . . . . . . . 10
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3 | 2 | eleq2d 2257 |
. . . . . . . . 9
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4 | 3 | anbi2d 464 |
. . . . . . . 8
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5 | 4 | rexbidv 2488 |
. . . . . . 7
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6 | 5 | orbi1d 792 |
. . . . . 6
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7 | 6 | abbidv 2305 |
. . . . 5
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8 | 7 | mpteq2dva 4105 |
. . . 4
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9 | recseq 6321 |
. . . 4
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10 | 8, 9 | syl 14 |
. . 3
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11 | 10 | reseq1d 4918 |
. 2
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12 | df-frec 6406 |
. 2
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13 | df-frec 6406 |
. 2
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14 | 11, 12, 13 | 3eqtr4g 2245 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-in 3147 df-uni 3822 df-br 4016 df-opab 4077 df-mpt 4078 df-res 4650 df-iota 5190 df-fv 5236 df-recs 6320 df-frec 6406 |
This theorem is referenced by: frecuzrdgdom 10432 frecuzrdgfun 10434 frecuzrdgsuct 10438 seqeq1 10462 seqeq2 10463 seqeq3 10464 iseqvalcbv 10471 hashfz1 10777 ennnfonelemr 12438 ctinfom 12443 isomninn 15133 iswomninn 15152 ismkvnn 15155 |
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