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

Theorem euabex 5472
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 2576 . 2 (∃!𝑥𝜑 → ∃*𝑥𝜑)
2 moabex 5470 . 2 (∃*𝑥𝜑 → {𝑥𝜑} ∈ V)
31, 2syl 17 1 (∃!𝑥𝜑 → {𝑥𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  ∃*wmo 2536  ∃!weu 2566  {cab 2712  Vcvv 3478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-un 3968  df-in 3970  df-ss 3980  df-sn 4632  df-pr 4634
This theorem is referenced by:  tfsconcatun  43327  sprval  47404  prprval  47439
  Copyright terms: Public domain W3C validator