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Theorem clelsb3 2715
Description: Substitution applied to an atomic wff (class version of elsb3 2421). (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 1829 . . 3 𝑦 𝑤𝐴
21sbco2 2402 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑤]𝑤𝐴)
3 nfv 1829 . . . 4 𝑤 𝑦𝐴
4 eleq1 2675 . . . 4 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
53, 4sbie 2395 . . 3 ([𝑦 / 𝑤]𝑤𝐴𝑦𝐴)
65sbbii 1873 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑦]𝑦𝐴)
7 nfv 1829 . . 3 𝑤 𝑥𝐴
8 eleq1 2675 . . 3 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
97, 8sbie 2395 . 2 ([𝑥 / 𝑤]𝑤𝐴𝑥𝐴)
102, 6, 93bitr3i 288 1 ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 194  [wsb 1866  wcel 1976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-cleq 2602  df-clel 2605
This theorem is referenced by:  hblem  2717  cbvreu  3144  sbcel1v  3461  rmo3  3493  kmlem15  8846  iuninc  28567  measiuns  29413  ballotlemodife  29692  bj-nfcf  31915  sbcel1gvOLD  37919  ellimcabssub0  38488
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