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Theorem clelsb3f 2924
 Description: Substitution applied to an atomic wff (class version of elsb3 2119). Usage of this theorem is discouraged because it depends on ax-13 2379. See clelsb3fw 2923 not requiring ax-13 2379, but extra disjoint variables. (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.) (New usage is discouraged.)
Hypothesis
Ref Expression
clelsb3f.1 𝑥𝐴
Assertion
Ref Expression
clelsb3f ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)

Proof of Theorem clelsb3f
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 clelsb3f.1 . . . 4 𝑥𝐴
21nfcri 2906 . . 3 𝑥 𝑤𝐴
32sbco2 2530 . 2 ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤𝐴 ↔ [𝑦 / 𝑤]𝑤𝐴)
4 clelsb3 2879 . . 3 ([𝑥 / 𝑤]𝑤𝐴𝑥𝐴)
54sbbii 2081 . 2 ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤𝐴 ↔ [𝑦 / 𝑥]𝑥𝐴)
6 clelsb3 2879 . 2 ([𝑦 / 𝑤]𝑤𝐴𝑦𝐴)
73, 5, 63bitr3i 304 1 ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  [wsb 2069   ∈ wcel 2111  Ⅎwnfc 2899 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 2113  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clel 2830  df-nfc 2901 This theorem is referenced by: (None)
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