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Theorem freceq2 6011
 Description: Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
Assertion
Ref Expression
freceq2 frec frec

Proof of Theorem freceq2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 106 . . . . . . . . 9
21eleq2d 2123 . . . . . . . 8
32anbi2d 445 . . . . . . 7
43orbi2d 714 . . . . . 6
54abbidv 2171 . . . . 5
65mpteq2dva 3875 . . . 4
7 recseq 5952 . . . 4 recs recs
86, 7syl 14 . . 3 recs recs
98reseq1d 4639 . 2 recs recs
10 df-frec 6009 . 2 frec recs
11 df-frec 6009 . 2 frec recs
129, 10, 113eqtr4g 2113 1 frec frec
 Colors of variables: wff set class Syntax hints:   wi 4   wa 101   wo 639   wceq 1259   wcel 1409  cab 2042  wrex 2324  cvv 2574  c0 3252   cmpt 3846   csuc 4130  com 4341   cdm 4373   cres 4375  cfv 4930  recscrecs 5950  freccfrec 6008 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2952  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-res 4385  df-iota 4895  df-fv 4938  df-recs 5951  df-frec 6009 This theorem is referenced by:  iseqeq1  9378  iseqeq3  9380
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