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Theorem clelsb3 2142
 Description: Substitution applied to an atomic wff (class version of elsb3 1852). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
clelsb3 ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
Distinct variable group:   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem clelsb3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfv 1421 . . 3 𝑦 𝑤𝐴
21sbco2 1839 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑤]𝑤𝐴)
3 nfv 1421 . . . 4 𝑤 𝑦𝐴
4 eleq1 2100 . . . 4 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
53, 4sbie 1674 . . 3 ([𝑦 / 𝑤]𝑤𝐴𝑦𝐴)
65sbbii 1648 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑦]𝑦𝐴)
7 nfv 1421 . . 3 𝑤 𝑥𝐴
8 eleq1 2100 . . 3 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
97, 8sbie 1674 . 2 ([𝑥 / 𝑤]𝑤𝐴𝑥𝐴)
102, 6, 93bitr3i 199 1 ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   ∈ wcel 1393  [wsb 1645 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-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036 This theorem is referenced by:  hblem  2145  nfraldya  2358  nfrexdya  2359  cbvreu  2531  sbcel1gv  2821  rmo3  2849  setindel  4263  elirr  4266  en2lp  4278  zfregfr  4298  tfi  4305  bdcriota  10003
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