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Theorem clelsb3 2716
 Description: Substitution applied to an atomic wff (class version of elsb3 2422). (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 1830 . . 3 𝑦 𝑤𝐴
21sbco2 2403 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑤]𝑤𝐴)
3 nfv 1830 . . . 4 𝑤 𝑦𝐴
4 eleq1 2676 . . . 4 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
53, 4sbie 2396 . . 3 ([𝑦 / 𝑤]𝑤𝐴𝑦𝐴)
65sbbii 1874 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑦]𝑦𝐴)
7 nfv 1830 . . 3 𝑤 𝑥𝐴
8 eleq1 2676 . . 3 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
97, 8sbie 2396 . 2 ([𝑥 / 𝑤]𝑤𝐴𝑥𝐴)
102, 6, 93bitr3i 289 1 ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  [wsb 1867   ∈ wcel 1977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-cleq 2603  df-clel 2606 This theorem is referenced by:  hblem  2718  cbvreu  3145  sbcel1v  3462  rmo3  3494  kmlem15  8869  iuninc  28761  measiuns  29607  ballotlemodife  29886  bj-nfcf  32112  sbcel1gvOLD  38116  ellimcabssub0  38684
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