Theorem rabidim2 45069

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

Assertion

Ref	Expression
rabidim2	⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝜑)

Proof of Theorem rabidim2

Step	Hyp	Ref	Expression
1		rabid 3424	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	simprbi 496	1 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝜑)

Syntax hints:   → wi 4   ∈ wcel 2109  {crab 3402

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 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403

This theorem is referenced by:  infnsuprnmpt  45217  preimagelt  46670  preimalegt  46671  pimrecltpos  46679  pimiooltgt  46681  pimrecltneg  46695  smfaddlem1  46734  smflimlem2  46743  smfrec  46760  smfmullem4  46765  smfdiv  46768  smfsupxr  46787  smfinflem  46788  smflimsuplem7  46797  smflimsuplem8  46798  fsupdm  46813  finfdm  46817