Theorem rabidim2 45878

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 3439	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	simprbi 503	1 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝜑)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  {crab 3418

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419

This theorem is used by:  infnsuprnmpt  46023  preimagelt  47471  preimalegt  47472  pimrecltpos  47480  pimiooltgt  47482  pimrecltneg  47496  sssmf  47510  smfaddlem1  47535  smflimlem2  47544  smfrec  47561  smfmullem4  47566  smfdiv  47569  smfsupxr  47588  smfinflem  47589  smflimsuplem7  47598  smflimsuplem8  47599  fsupdm  47614  finfdm  47618