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

Theorem euabex 5330
Description: The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
Assertion
Ref Expression
euabex (∃!𝑥𝜑 → {𝑥𝜑} ∈ V)

Proof of Theorem euabex
StepHypRef Expression
1 eumo 2662 . 2 (∃!𝑥𝜑 → ∃*𝑥𝜑)
2 moabex 5328 . 2 (∃*𝑥𝜑 → {𝑥𝜑} ∈ V)
31, 2syl 17 1 (∃!𝑥𝜑 → {𝑥𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  ∃*wmo 2620  ∃!weu 2652  {cab 2800  Vcvv 3469
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
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 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-sn 4540  df-pr 4542
This theorem is referenced by:  sprval  43936  prprval  43971
  Copyright terms: Public domain W3C validator