Theorem rabidim1 3421

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 3420	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	simplbi 497	1 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2113  {crab 3399

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-12 2184  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400

This theorem is referenced by:  frgrwopreglem5  30398  frgrwopreg  30400  rabexgfGS  32576  ssrab2f  45382  infnsuprnmpt  45515  preimagelt  46964  preimalegt  46965  pimrecltpos  46973  pimiooltgt  46975  pimrecltneg  46989  smfresal  47053  smfpimbor1lem2  47064  smflimmpt  47075  smfsupmpt  47080  smfinfmpt  47084  smflimsuplem7  47091  smflimsuplem8  47092  smflimsupmpt  47094  smfliminfmpt  47097  fsupdm  47107  finfdm  47111