Theorem rabidim1 3452

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 3451	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	simplbi 497	1 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2105  {crab 3431

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432

This theorem is referenced by:  frgrwopreglem5  29842  frgrwopreg  29844  rabexgfGS  32007  ssrab2f  44108  infnsuprnmpt  44253  preimagelt  45714  preimalegt  45715  pimrecltpos  45723  pimrecltneg  45739  smfresal  45803  smfpimbor1lem2  45814  smflimmpt  45825  smfsupmpt  45830  smfinfmpt  45834  smflimsuplem7  45841  smflimsuplem8  45842  smflimsupmpt  45844  smfliminfmpt  45847  fsupdm  45857  finfdm  45861