Theorem rabidim1 3297

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 3295	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	simplbi 492	1 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2157  {crab 3091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-12 2213  ax-ext 2775

This theorem depends on definitions:  df-bi 199  df-an 386  df-tru 1657  df-ex 1876  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-rab 3096

This theorem is referenced by:  frgrwopreglem5  27662  frgrwopreg  27664  ssrab2f  40046  infnsuprnmpt  40200  pimrecltpos  41653  pimrecltneg  41667  smfresal  41729  smfpimbor1lem2  41740  smflimmpt  41750  smfsupmpt  41755  smfinfmpt  41759  smflimsuplem7  41766  smflimsuplem8  41767  smflimsupmpt  41769  smfliminfmpt  41772