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Theorem elrab2w 3589
Description: Membership in a restricted class abstraction. This is to elrab2 3588 what elab2gw 3569 is to elab2g 3572. (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 3415 . 2 (𝐴𝐶𝐴 ∈ V)
2 elex 3415 . . 3 (𝐴𝐵𝐴 ∈ V)
32adantr 484 . 2 ((𝐴𝐵𝜒) → 𝐴 ∈ V)
4 eleq1w 2815 . . . 4 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
5 elrab2w.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
64, 5anbi12d 634 . . 3 (𝑥 = 𝑦 → ((𝑥𝐵𝜑) ↔ (𝑦𝐵𝜓)))
7 eleq1 2820 . . . 4 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
8 elrab2w.2 . . . 4 (𝑦 = 𝐴 → (𝜓𝜒))
97, 8anbi12d 634 . . 3 (𝑦 = 𝐴 → ((𝑦𝐵𝜓) ↔ (𝐴𝐵𝜒)))
10 elrab2w.3 . . . 4 𝐶 = {𝑥𝐵𝜑}
11 df-rab 3062 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
1210, 11eqtri 2761 . . 3 𝐶 = {𝑥 ∣ (𝑥𝐵𝜑)}
136, 9, 12elab2gw 3569 . 2 (𝐴 ∈ V → (𝐴𝐶 ↔ (𝐴𝐵𝜒)))
141, 3, 13pm5.21nii 383 1 (𝐴𝐶 ↔ (𝐴𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2113  {cab 2716  {crab 3057  Vcvv 3397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-rab 3062  df-v 3399
This theorem is referenced by:  elom  7596
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