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Theorem elin 3123
Description: Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.)
Assertion
Ref Expression
elin (A (B𝐶) ↔ (A B A 𝐶))

Proof of Theorem elin
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elex 2563 . 2 (A (B𝐶) → A V)
2 elex 2563 . . 3 (A 𝐶A V)
32adantl 262 . 2 ((A B A 𝐶) → A V)
4 eleq1 2100 . . . 4 (x = A → (x BA B))
5 eleq1 2100 . . . 4 (x = A → (x 𝐶A 𝐶))
64, 5anbi12d 442 . . 3 (x = A → ((x B x 𝐶) ↔ (A B A 𝐶)))
7 df-in 2921 . . 3 (B𝐶) = {x ∣ (x B x 𝐶)}
86, 7elab2g 2686 . 2 (A V → (A (B𝐶) ↔ (A B A 𝐶)))
91, 3, 8pm5.21nii 620 1 (A (B𝐶) ↔ (A B A 𝐶))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1243   wcel 1393  Vcvv 2554  cin 2913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-in 2921
This theorem is referenced by:  elin2  3124  elin3  3125  incom  3126  ineqri  3127  ineq1  3128  inass  3144  inss1  3154  ssin  3156  ssrin  3159  inssdif  3170  difin  3171  unssin  3173  inssun  3174  invdif  3176  indif  3177  indi  3181  undi  3182  difundi  3186  difindiss  3188  indifdir  3190  difin2  3196  inrab2  3207  inelcm  3279  inssdif0im  3288  uniin  3594  intun  3640  intpr  3641  elrint  3649  iunin2  3714  iinin2m  3719  elriin  3721  brin  3805  trin  3858  inex1  3885  inuni  3903  bnd2  3920  ordpwsucss  4246  ordpwsucexmid  4249  peano5  4267  inopab  4414  inxp  4416  dmin  4489  opelres  4563  intasym  4655  asymref  4656  dminss  4684  imainss  4685  inimasn  4687  ssrnres  4709  cnvresima  4756  dfco2a  4767  imainlem  4926  imain  4927  2elresin  4956  nfvres  5152  respreima  5241  isoini  5403  offval  5664  tfrlem5  5875  peano5nnnn  6856  peano5nni  7790  ixxdisj  8634  icodisj  8722  fzdisj  8778  uzdisj  8817  nn0disj  8857  fzouzdisj  8898  bdinex1  9462  bj-indind  9499  peano5setOLD  9508
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