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Theorem elrabiOLD 3673
Description: Obsolete version of elrabi 3672 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 2873 . . 3 (𝐴 ∈ {𝑥 ∣ (𝑥𝑉𝜑)} ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝑉𝜑)))
2 eleq1 2815 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑉𝐴𝑉))
32anbi1d 629 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝑉𝜑) ↔ (𝐴𝑉𝜑)))
43simprbda 498 . . . 4 ((𝑥 = 𝐴 ∧ (𝑥𝑉𝜑)) → 𝐴𝑉)
54exlimiv 1925 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝑉𝜑)) → 𝐴𝑉)
61, 5sylbi 216 . 2 (𝐴 ∈ {𝑥 ∣ (𝑥𝑉𝜑)} → 𝐴𝑉)
7 df-rab 3427 . 2 {𝑥𝑉𝜑} = {𝑥 ∣ (𝑥𝑉𝜑)}
86, 7eleq2s 2845 1 (𝐴 ∈ {𝑥𝑉𝜑} → 𝐴𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wex 1773  wcel 2098  {cab 2703  {crab 3426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427
This theorem is referenced by: (None)
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