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Theorem clelsb3fw 2962
Description: Substitution applied to an atomic wff (class version of elsb3 2120). Version of clelsb3f 2963 with a disjoint variable condition, which does not require ax-13 2382. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypothesis
Ref Expression
clelsb3fw.1 𝑥𝐴
Assertion
Ref Expression
clelsb3fw ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem clelsb3fw
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 clelsb3fw.1 . . . 4 𝑥𝐴
21nfcri 2946 . . 3 𝑥 𝑤𝐴
32sbco2v 2344 . 2 ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤𝐴 ↔ [𝑦 / 𝑤]𝑤𝐴)
4 clelsb3 2920 . . 3 ([𝑥 / 𝑤]𝑤𝐴𝑥𝐴)
54sbbii 2081 . 2 ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤𝐴 ↔ [𝑦 / 𝑥]𝑥𝐴)
6 clelsb3 2920 . 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 2112  wnfc 2939
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 2114  ax-10 2143  ax-11 2159  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2070  df-clel 2873  df-nfc 2941
This theorem is referenced by:  rmo3f  3676  suppss2f  30401  fmptdF  30422  disjdsct  30465  esumpfinvalf  31443
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