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Theorem rabidim1 3256
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 3254 . 2 (𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
21simplbi 478 1 (𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2139  {crab 3054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-12 2196  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1635  df-ex 1854  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-rab 3059
This theorem is referenced by:  frgrwopreglem5  27475  frgrwopreg  27477  ssrab2f  39799  infnsuprnmpt  39964  pimrecltpos  41425  pimrecltneg  41439  smfresal  41501  smfpimbor1lem2  41512  smflimmpt  41522  smfsupmpt  41527  smfinfmpt  41531  smflimsuplem7  41538  smflimsuplem8  41539  smflimsupmpt  41541  smfliminfmpt  41544
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