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Theorem clelsb3 2732
Description: Substitution applied to an atomic wff (class version of elsb3 2438). (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 1845 . . 3 𝑦 𝑤𝐴
21sbco2 2419 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑤]𝑤𝐴)
3 nfv 1845 . . . 4 𝑤 𝑦𝐴
4 eleq1 2692 . . . 4 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
53, 4sbie 2412 . . 3 ([𝑦 / 𝑤]𝑤𝐴𝑦𝐴)
65sbbii 1889 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑦]𝑦𝐴)
7 nfv 1845 . . 3 𝑤 𝑥𝐴
8 eleq1 2692 . . 3 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
97, 8sbie 2412 . 2 ([𝑥 / 𝑤]𝑤𝐴𝑥𝐴)
102, 6, 93bitr3i 290 1 ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  [wsb 1882  wcel 1992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-cleq 2619  df-clel 2622
This theorem is referenced by:  hblem  2734  cbvreu  3162  sbcel1v  3482  rmo3  3514  kmlem15  8931  iuninc  29215  measiuns  30053  ballotlemodife  30332  bj-nfcf  32540  sbcel1gvOLD  38544  ellimcabssub0  39221
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