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Theorem clelsb3 2941
 Description: Substitution applied to an atomic wff (class version of elsb3 2122). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
clelsb3 ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem clelsb3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eleq1w 2896 . 2 (𝑥 = 𝑤 → (𝑥𝐴𝑤𝐴))
2 eleq1w 2896 . 2 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
31, 2sbievw2 2107 1 ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  [wsb 2069   ∈ wcel 2114 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clel 2894 This theorem is referenced by:  hblem  2944  hblemg  2945  abbi2dv  2951  nfcriiOLD  2973  clelsb3fw  2983  clelsb3f  2984  cbvreuw  3417  cbvreu  3422  sbcel1v  3813  rmo3  3845  kmlem15  9579  iuninc  30319  measiuns  31550  ballotlemodife  31829  bj-nfcf  34327  wl-dfrabv  34989  wl-clelsb3df  34990  wl-dfrabf  34991  ellimcabssub0  42202
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