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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 2209 | . . . . . . . 8 |
3 | 2 | anbi2d 459 | . . . . . . 7 |
4 | 3 | orbi2d 779 | . . . . . 6 |
5 | 4 | abbidv 2257 | . . . . 5 |
6 | 5 | mpteq2dva 4018 | . . . 4 |
7 | recseq 6203 | . . . 4 recs recs | |
8 | 6, 7 | syl 14 | . . 3 recs recs |
9 | 8 | reseq1d 4818 | . 2 recs recs |
10 | df-frec 6288 | . 2 frec recs | |
11 | df-frec 6288 | . 2 frec recs | |
12 | 9, 10, 11 | 3eqtr4g 2197 | 1 frec frec |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 wceq 1331 wcel 1480 cab 2125 wrex 2417 cvv 2686 c0 3363 cmpt 3989 csuc 4287 com 4504 cdm 4539 cres 4541 cfv 5123 recscrecs 6201 freccfrec 6287 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-in 3077 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-res 4551 df-iota 5088 df-fv 5131 df-recs 6202 df-frec 6288 |
This theorem is referenced by: seqeq1 10221 seqeq3 10223 |
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