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

Theorem absneu 4638
Description: Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
Assertion
Ref Expression
absneu ((𝐴𝑉 ∧ {𝑥𝜑} = {𝐴}) → ∃!𝑥𝜑)

Proof of Theorem absneu
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sneq 4549 . . . . 5 (𝑦 = 𝐴 → {𝑦} = {𝐴})
21eqeq2d 2833 . . . 4 (𝑦 = 𝐴 → ({𝑥𝜑} = {𝑦} ↔ {𝑥𝜑} = {𝐴}))
32spcegv 3572 . . 3 (𝐴𝑉 → ({𝑥𝜑} = {𝐴} → ∃𝑦{𝑥𝜑} = {𝑦}))
43imp 410 . 2 ((𝐴𝑉 ∧ {𝑥𝜑} = {𝐴}) → ∃𝑦{𝑥𝜑} = {𝑦})
5 euabsn2 4635 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
64, 5sylibr 237 1 ((𝐴𝑉 ∧ {𝑥𝜑} = {𝐴}) → ∃!𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2114  ∃!weu 2652  {cab 2800  {csn 4539
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
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-sn 4540
This theorem is referenced by:  rabsneu  4639
  Copyright terms: Public domain W3C validator