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Theorem rabidim1 3378
Description: Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Assertion
Ref Expression
rabidim1 (𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴)

Proof of Theorem rabidim1
StepHypRef Expression
1 rabid 3376 . 2 (𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
21simplbi 498 1 (𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  {crab 3139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1531  df-ex 1772  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-rab 3144
This theorem is referenced by:  frgrwopreglem5  28027  frgrwopreg  28029  rabexgfGS  30189  ssrab2f  41260  infnsuprnmpt  41398  preimagelt  42857  preimalegt  42858  pimrecltpos  42864  pimrecltneg  42878  smfresal  42940  smfpimbor1lem2  42951  smflimmpt  42961  smfsupmpt  42966  smfinfmpt  42970  smflimsuplem7  42977  smflimsuplem8  42978  smflimsupmpt  42980  smfliminfmpt  42983
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