Theorem rabidim1 3431

Description: Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Assertion

Ref	Expression
rabidim1	⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)

Proof of Theorem rabidim1

Step	Hyp	Ref	Expression
1		rabid 3430	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	simplbi 497	1 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2109  {crab 3408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409

This theorem is referenced by:  frgrwopreglem5  30257  frgrwopreg  30259  rabexgfGS  32435  ssrab2f  45118  infnsuprnmpt  45251  preimagelt  46704  preimalegt  46705  pimrecltpos  46713  pimrecltneg  46729  smfresal  46793  smfpimbor1lem2  46804  smflimmpt  46815  smfsupmpt  46820  smfinfmpt  46824  smflimsuplem7  46831  smflimsuplem8  46832  smflimsupmpt  46834  smfliminfmpt  46837  fsupdm  46847  finfdm  46851