Theorem rabidim1 3380

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 3378	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	simplbi 500	1 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2114  {crab 3142

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1540  df-ex 1781  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-rab 3147

This theorem is referenced by:  frgrwopreglem5  28100  frgrwopreg  28102  rabexgfGS  30262  ssrab2f  41403  infnsuprnmpt  41542  preimagelt  43000  preimalegt  43001  pimrecltpos  43007  pimrecltneg  43021  smfresal  43083  smfpimbor1lem2  43094  smflimmpt  43104  smfsupmpt  43109  smfinfmpt  43113  smflimsuplem7  43120  smflimsuplem8  43121  smflimsupmpt  43123  smfliminfmpt  43126