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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 2240 | . . . . . . . 8 |
3 | 2 | anbi2d 461 | . . . . . . 7 |
4 | 3 | orbi2d 785 | . . . . . 6 |
5 | 4 | abbidv 2288 | . . . . 5 |
6 | 5 | mpteq2dva 4079 | . . . 4 |
7 | recseq 6285 | . . . 4 recs recs | |
8 | 6, 7 | syl 14 | . . 3 recs recs |
9 | 8 | reseq1d 4890 | . 2 recs recs |
10 | df-frec 6370 | . 2 frec recs | |
11 | df-frec 6370 | . 2 frec recs | |
12 | 9, 10, 11 | 3eqtr4g 2228 | 1 frec frec |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 703 wceq 1348 wcel 2141 cab 2156 wrex 2449 cvv 2730 c0 3414 cmpt 4050 csuc 4350 com 4574 cdm 4611 cres 4613 cfv 5198 recscrecs 6283 freccfrec 6369 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-in 3127 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-res 4623 df-iota 5160 df-fv 5206 df-recs 6284 df-frec 6370 |
This theorem is referenced by: seqeq1 10404 seqeq3 10406 |
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