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Theorem absneu 4725
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 4631 . . . . 5 (𝑦 = 𝐴 → {𝑦} = {𝐴})
21eqeq2d 2735 . . . 4 (𝑦 = 𝐴 → ({𝑥𝜑} = {𝑦} ↔ {𝑥𝜑} = {𝐴}))
32spcegv 3579 . . 3 (𝐴𝑉 → ({𝑥𝜑} = {𝐴} → ∃𝑦{𝑥𝜑} = {𝑦}))
43imp 406 . 2 ((𝐴𝑉 ∧ {𝑥𝜑} = {𝐴}) → ∃𝑦{𝑥𝜑} = {𝑦})
5 euabsn2 4722 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
64, 5sylibr 233 1 ((𝐴𝑉 ∧ {𝑥𝜑} = {𝐴}) → ∃!𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wex 1773  wcel 2098  ∃!weu 2554  {cab 2701  {csn 4621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-sn 4622
This theorem is referenced by:  rabsneu  4726
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