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Theorem ndxid 12042
 Description: A structure component extractor is defined by its own index. This theorem, together with strslfv 12062 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the in df-base 12024, making it easier to change should the need arise. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.) (Proof shortened by BJ, 27-Dec-2021.)
Hypotheses
Ref Expression
ndxarg.1 Slot
ndxarg.2
Assertion
Ref Expression
ndxid Slot

Proof of Theorem ndxid
StepHypRef Expression
1 ndxarg.1 . . . 4 Slot
2 ndxarg.2 . . . 4
31, 2ndxarg 12041 . . 3
43eqcomi 2144 . 2
5 sloteq 12023 . . 3 Slot Slot
61, 5syl5eq 2185 . 2 Slot
74, 6ax-mp 5 1 Slot
 Colors of variables: wff set class Syntax hints:   wceq 1332   wcel 1481  cfv 5132  cn 8764  cnx 12015  Slot cslot 12017 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140  ax-un 4364  ax-cnex 7755  ax-resscn 7756  ax-1re 7758  ax-addrcl 7761 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-sbc 2915  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-int 3781  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-iota 5097  df-fun 5134  df-fv 5140  df-inn 8765  df-ndx 12021  df-slot 12022 This theorem is referenced by:  ndxslid  12043  strndxid  12046  baseid  12071  plusgid  12112  mulrid  12129  starvid  12138  scaid  12146  vscaid  12149  ipid  12157  tsetid  12167  pleid  12174  dsid  12177
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