Theorem rabidim1 3454

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 3453	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	simplbi 499	1 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2107  {crab 3433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434

This theorem is referenced by:  frgrwopreglem5  29574  frgrwopreg  29576  rabexgfGS  31739  ssrab2f  43806  infnsuprnmpt  43954  preimagelt  45415  preimalegt  45416  pimrecltpos  45424  pimrecltneg  45440  smfresal  45504  smfpimbor1lem2  45515  smflimmpt  45526  smfsupmpt  45531  smfinfmpt  45535  smflimsuplem7  45542  smflimsuplem8  45543  smflimsupmpt  45545  smfliminfmpt  45548  fsupdm  45558  finfdm  45562