Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rmorabex Structured version   Visualization version   GIF version

Theorem rmorabex 5077
 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 5076 . 2 (∃*𝑥(𝑥𝐴𝜑) → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
2 df-rmo 3058 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
3 df-rab 3059 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
43eleq1i 2830 . 2 ({𝑥𝐴𝜑} ∈ V ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
51, 2, 43imtr4i 281 1 (∃*𝑥𝐴 𝜑 → {𝑥𝐴𝜑} ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 2139  ∃*wmo 2608  {cab 2746  ∃*wrmo 3053  {crab 3054  Vcvv 3340 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rmo 3058  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-sn 4322  df-pr 4324 This theorem is referenced by:  supexd  8524
 Copyright terms: Public domain W3C validator