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Theorem clelsb3 2888
 Description: Substitution applied to an atomic wff (class version of elsb3 2513). (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 1957 . . 3 𝑦 𝑤𝐴
21sbco2 2492 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑤]𝑤𝐴)
3 nfv 1957 . . . 4 𝑤 𝑦𝐴
4 eleq1w 2842 . . . 4 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
53, 4sbie 2484 . . 3 ([𝑦 / 𝑤]𝑤𝐴𝑦𝐴)
65sbbii 2019 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑦]𝑦𝐴)
7 nfv 1957 . . 3 𝑤 𝑥𝐴
8 eleq1w 2842 . . 3 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
97, 8sbie 2484 . 2 ([𝑥 / 𝑤]𝑤𝐴𝑥𝐴)
102, 6, 93bitr3i 293 1 ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198  [wsb 2011   ∈ wcel 2107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clel 2774 This theorem is referenced by:  hblem  2891  clelsb3f  2938  cbvreu  3365  sbcel1vOLD  3714  rmo3OLD  3746  iuninc  29941  measiuns  30878  ballotlemodife  31158  bj-nfcf  33493  wl-clelsb3df  33998  ellimcabssub0  40761
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