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Theorem clelsb3f 2951
Description: Substitution applied to an atomic wff (class version of elsb3 2597). (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 2941 . . 3 𝑦 𝑤𝐴
32sbco2 2576 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑤]𝑤𝐴)
4 nfv 2008 . . . 4 𝑤 𝑦𝐴
5 eleq1w 2867 . . . 4 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
64, 5sbie 2569 . . 3 ([𝑦 / 𝑤]𝑤𝐴𝑦𝐴)
76sbbii 2069 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑦]𝑦𝐴)
8 nfv 2008 . . 3 𝑤 𝑥𝐴
9 eleq1w 2867 . . 3 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
108, 9sbie 2569 . 2 ([𝑥 / 𝑤]𝑤𝐴𝑥𝐴)
113, 7, 103bitr3i 292 1 ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 197  [wsb 2062  wcel 2158  wnfc 2934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-clel 2801  df-nfc 2936
This theorem is referenced by:  rmo3f  29658  suppss2f  29763  fmptdF  29780  disjdsct  29804  esumpfinvalf  30460
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