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Theorem clelsb1f 2907
Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2113). Usage of this theorem is discouraged because it depends on ax-13 2370. See clelsb1fw 2906 not requiring ax-13 2370, 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
clelsb1f.1 𝑥𝐴
Assertion
Ref Expression
clelsb1f ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)

Proof of Theorem clelsb1f
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 clelsb1f.1 . . . 4 𝑥𝐴
21nfcri 2889 . . 3 𝑥 𝑤𝐴
32sbco2 2509 . 2 ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤𝐴 ↔ [𝑦 / 𝑤]𝑤𝐴)
4 clelsb1 2859 . . 3 ([𝑥 / 𝑤]𝑤𝐴𝑥𝐴)
54sbbii 2078 . 2 ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤𝐴 ↔ [𝑦 / 𝑥]𝑥𝐴)
6 clelsb1 2859 . 2 ([𝑦 / 𝑤]𝑤𝐴𝑦𝐴)
73, 5, 63bitr3i 301 1 ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  [wsb 2066  wcel 2105  wnfc 2882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-10 2136  ax-11 2153  ax-12 2170  ax-13 2370
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clel 2809  df-nfc 2884
This theorem is referenced by: (None)
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