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Theorem clelsb3f 29292
Description: Substitution applied to an atomic wff (class version of elsb3 2432). (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 2756 . . 3 𝑦 𝑤𝐴
32sbco2 2413 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑤]𝑤𝐴)
4 nfv 1841 . . . 4 𝑤 𝑦𝐴
5 eleq1 2687 . . . 4 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
64, 5sbie 2406 . . 3 ([𝑦 / 𝑤]𝑤𝐴𝑦𝐴)
76sbbii 1885 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑦]𝑦𝐴)
8 nfv 1841 . . 3 𝑤 𝑥𝐴
9 eleq1 2687 . . 3 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
108, 9sbie 2406 . 2 ([𝑥 / 𝑤]𝑤𝐴𝑥𝐴)
113, 7, 103bitr3i 290 1 ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  [wsb 1878  wcel 1988  wnfc 2749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-cleq 2613  df-clel 2616  df-nfc 2751
This theorem is referenced by:  rmo3f  29307  suppss2f  29412  fmptdF  29429  disjdsct  29454  esumpfinvalf  30112
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