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Theorem elrabiOLD 3610
Description: Obsolete version of elrabi 3609 as of 5-Aug-2024. (Contributed by Alexander van der Vekens, 31-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
elrabiOLD (𝐴 ∈ {𝑥𝑉𝜑} → 𝐴𝑉)
Distinct variable groups:   𝑥,𝑉   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elrabiOLD
StepHypRef Expression
1 clelab 2881 . . 3 (𝐴 ∈ {𝑥 ∣ (𝑥𝑉𝜑)} ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝑉𝜑)))
2 eleq1 2826 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑉𝐴𝑉))
32anbi1d 633 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝑉𝜑) ↔ (𝐴𝑉𝜑)))
43simprbda 502 . . . 4 ((𝑥 = 𝐴 ∧ (𝑥𝑉𝜑)) → 𝐴𝑉)
54exlimiv 1938 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝑉𝜑)) → 𝐴𝑉)
61, 5sylbi 220 . 2 (𝐴 ∈ {𝑥 ∣ (𝑥𝑉𝜑)} → 𝐴𝑉)
7 df-rab 3071 . 2 {𝑥𝑉𝜑} = {𝑥 ∣ (𝑥𝑉𝜑)}
86, 7eleq2s 2857 1 (𝐴 ∈ {𝑥𝑉𝜑} → 𝐴𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wex 1787  wcel 2111  {cab 2715  {crab 3066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2176  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3071
This theorem is referenced by: (None)
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