Users' Mathboxes Mathbox for Steven Nguyen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elrab2w Structured version   Visualization version   GIF version

Theorem elrab2w 40167
Description: Membership in a restricted class abstraction. This is to elrab2 3627 what elab2gw 40166 is to elab2g 3611. (Contributed by SN, 2-Sep-2024.)
Hypotheses
Ref Expression
elrab2w.1 (𝑥 = 𝑦 → (𝜑𝜓))
elrab2w.2 (𝑦 = 𝐴 → (𝜓𝜒))
elrab2w.3 𝐶 = {𝑥𝐵𝜑}
Assertion
Ref Expression
elrab2w (𝐴𝐶 ↔ (𝐴𝐵𝜒))
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑦   𝜒,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑥)   𝐴(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem elrab2w
StepHypRef Expression
1 elex 3450 . 2 (𝐴𝐶𝐴 ∈ V)
2 elex 3450 . . 3 (𝐴𝐵𝐴 ∈ V)
32adantr 481 . 2 ((𝐴𝐵𝜒) → 𝐴 ∈ V)
4 eleq1w 2821 . . . 4 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
5 elrab2w.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
64, 5anbi12d 631 . . 3 (𝑥 = 𝑦 → ((𝑥𝐵𝜑) ↔ (𝑦𝐵𝜓)))
7 eleq1 2826 . . . 4 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
8 elrab2w.2 . . . 4 (𝑦 = 𝐴 → (𝜓𝜒))
97, 8anbi12d 631 . . 3 (𝑦 = 𝐴 → ((𝑦𝐵𝜓) ↔ (𝐴𝐵𝜒)))
10 elrab2w.3 . . . 4 𝐶 = {𝑥𝐵𝜑}
11 df-rab 3073 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
1210, 11eqtri 2766 . . 3 𝐶 = {𝑥 ∣ (𝑥𝐵𝜑)}
136, 9, 12elab2gw 40166 . 2 (𝐴 ∈ V → (𝐴𝐶 ↔ (𝐴𝐵𝜒)))
141, 3, 13pm5.21nii 380 1 (𝐴𝐶 ↔ (𝐴𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  {cab 2715  {crab 3068  Vcvv 3432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator