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Theorem clelsb3f 2936
Description: Substitution applied to an atomic wff (class version of elsb3 2408). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.) (Proof shortened by Wolf Lammen, 7-May-2023.)
Hypothesis
Ref Expression
clelsb3f.1 𝑦𝐴
Assertion
Ref Expression
clelsb3f ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)

Proof of Theorem clelsb3f
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 clelsb3f.1 . . . 4 𝑦𝐴
21nfcri 2926 . . 3 𝑦 𝑤𝐴
32sbco2 2477 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑤]𝑤𝐴)
4 clelsb3 2893 . . 3 ([𝑦 / 𝑤]𝑤𝐴𝑦𝐴)
54sbbii 2027 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑦]𝑦𝐴)
6 clelsb3 2893 . 2 ([𝑥 / 𝑤]𝑤𝐴𝑥𝐴)
73, 5, 63bitr3i 293 1 ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 198  [wsb 2015  wcel 2050  wnfc 2916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clel 2846  df-nfc 2918
This theorem is referenced by:  rmo3f  3637  suppss2f  30146  fmptdF  30163  disjdsct  30197  esumpfinvalf  30985
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