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Theorem rmorabex 5086
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 5085 . 2 (∃*𝑥(𝑥𝐴𝜑) → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
2 df-rmo 3063 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
3 df-rab 3064 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
43eleq1i 2835 . 2 ({𝑥𝐴𝜑} ∈ V ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
51, 2, 43imtr4i 283 1 (∃*𝑥𝐴 𝜑 → {𝑥𝐴𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 2155  ∃*wmo 2563  {cab 2751  ∃*wrmo 3058  {crab 3059  Vcvv 3350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rmo 3063  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-sn 4337  df-pr 4339
This theorem is referenced by:  supexd  8570
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