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Mirrors > Home > ILE Home > Th. List > freceq2 | Unicode version |
Description: Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
Ref | Expression |
---|---|
freceq2 | frec frec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . . . . . . 9 | |
2 | 1 | eleq2d 2227 | . . . . . . . 8 |
3 | 2 | anbi2d 460 | . . . . . . 7 |
4 | 3 | orbi2d 780 | . . . . . 6 |
5 | 4 | abbidv 2275 | . . . . 5 |
6 | 5 | mpteq2dva 4054 | . . . 4 |
7 | recseq 6253 | . . . 4 recs recs | |
8 | 6, 7 | syl 14 | . . 3 recs recs |
9 | 8 | reseq1d 4865 | . 2 recs recs |
10 | df-frec 6338 | . 2 frec recs | |
11 | df-frec 6338 | . 2 frec recs | |
12 | 9, 10, 11 | 3eqtr4g 2215 | 1 frec frec |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 698 wceq 1335 wcel 2128 cab 2143 wrex 2436 cvv 2712 c0 3394 cmpt 4025 csuc 4325 com 4549 cdm 4586 cres 4588 cfv 5170 recscrecs 6251 freccfrec 6337 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-in 3108 df-uni 3773 df-br 3966 df-opab 4026 df-mpt 4027 df-res 4598 df-iota 5135 df-fv 5178 df-recs 6252 df-frec 6338 |
This theorem is referenced by: seqeq1 10347 seqeq3 10349 |
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