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Theorem rmorabex 5340
Description: Restricted "at most one" existence implies a restricted class abstraction exists. (Contributed by NM, 17-Jun-2017.)
Assertion
Ref Expression
rmorabex (∃*𝑥𝐴 𝜑 → {𝑥𝐴𝜑} ∈ V)

Proof of Theorem rmorabex
StepHypRef Expression
1 moabex 5339 . 2 (∃*𝑥(𝑥𝐴𝜑) → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
2 df-rmo 3141 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
3 df-rab 3142 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
43eleq1i 2906 . 2 ({𝑥𝐴𝜑} ∈ V ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
51, 2, 43imtr4i 295 1 (∃*𝑥𝐴 𝜑 → {𝑥𝐴𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  ∃*wmo 2622  {cab 2802  ∃*wrmo 3136  {crab 3137  Vcvv 3480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rmo 3141  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553
This theorem is referenced by:  supexd  8916
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