Theorem rabidim2 45288

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 3418	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	simprbi 496	1 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝜑)

Syntax hints:   → wi 4   ∈ wcel 2113  {crab 3397

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 2182  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398

This theorem is referenced by:  infnsuprnmpt  45436  preimagelt  46885  preimalegt  46886  pimrecltpos  46894  pimiooltgt  46896  pimrecltneg  46910  smfaddlem1  46949  smflimlem2  46958  smfrec  46975  smfmullem4  46980  smfdiv  46983  smfsupxr  47002  smfinflem  47003  smflimsuplem7  47012  smflimsuplem8  47013  fsupdm  47028  finfdm  47032