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Theorem absneu 4733
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 4641 . . . . 5 (𝑦 = 𝐴 → {𝑦} = {𝐴})
21eqeq2d 2746 . . . 4 (𝑦 = 𝐴 → ({𝑥𝜑} = {𝑦} ↔ {𝑥𝜑} = {𝐴}))
32spcegv 3597 . . 3 (𝐴𝑉 → ({𝑥𝜑} = {𝐴} → ∃𝑦{𝑥𝜑} = {𝑦}))
43imp 406 . 2 ((𝐴𝑉 ∧ {𝑥𝜑} = {𝐴}) → ∃𝑦{𝑥𝜑} = {𝑦})
5 euabsn2 4730 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
64, 5sylibr 234 1 ((𝐴𝑉 ∧ {𝑥𝜑} = {𝐴}) → ∃!𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  ∃!weu 2566  {cab 2712  {csn 4631
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
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-sn 4632
This theorem is referenced by:  rabsneu  4734
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