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Theorem clelsb3f 28522
Description: Substitution applied to an atomic wff (class version of elsb3 2326). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
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 2649 . . 3 𝑦 𝑤𝐴
32sbco2 2307 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑤]𝑤𝐴)
4 nfv 1796 . . . 4 𝑤 𝑦𝐴
5 eleq1 2580 . . . 4 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
64, 5sbie 2300 . . 3 ([𝑦 / 𝑤]𝑤𝐴𝑦𝐴)
76sbbii 1837 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑦]𝑦𝐴)
8 nfv 1796 . . 3 𝑤 𝑥𝐴
9 eleq1 2580 . . 3 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
108, 9sbie 2300 . 2 ([𝑥 / 𝑤]𝑤𝐴𝑥𝐴)
113, 7, 103bitr3i 288 1 ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 194  [wsb 1830  wcel 1938  wnfc 2642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ex 1695  df-nf 1699  df-sb 1831  df-cleq 2507  df-clel 2510  df-nfc 2644
This theorem is referenced by:  rmo3f  28537  suppss2f  28637  fmptdF  28654  disjdsct  28681  esumpfinvalf  29291
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