Theorem rabidim1 3449

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 3448	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	simplbi 497	1 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2099  {crab 3428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3429

This theorem is referenced by:  frgrwopreglem5  30124  frgrwopreg  30126  rabexgfGS  32290  ssrab2f  44477  infnsuprnmpt  44620  preimagelt  46081  preimalegt  46082  pimrecltpos  46090  pimrecltneg  46106  smfresal  46170  smfpimbor1lem2  46181  smflimmpt  46192  smfsupmpt  46197  smfinfmpt  46201  smflimsuplem7  46208  smflimsuplem8  46209  smflimsupmpt  46211  smfliminfmpt  46214  fsupdm  46224  finfdm  46228