Theorem rabidim1 3437

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 3436	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	simplbi 502	1 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  {crab 3416

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417

This theorem is used by:  frgrwopreglem5  30949  frgrwopreg  30951  rabexgfGS  33164  ssrab2f  46335  infnsuprnmpt  46465  preimagelt  47912  preimalegt  47913  pimrecltpos  47921  pimiooltgt  47923  pimrecltneg  47937  smfresal  48001  smfpimbor1lem2  48012  smflimmpt  48023  smfsupmpt  48028  smfinfmpt  48032  smflimsuplem7  48039  smflimsuplem8  48040  smflimsupmpt  48042  smfliminfmpt  48045  fsupdm  48055  finfdm  48059