Theorem absneu 4692

Description: Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)

Assertion

Ref	Expression
absneu	⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ 𝜑} = {𝐴}) → ∃!𝑥𝜑)

Proof of Theorem absneu

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4597	. . . . 5 ⊢ (𝑦 = 𝐴 → {𝑦} = {𝐴})
2	1	eqeq2d 2773	. . . 4 ⊢ (𝑦 = 𝐴 → ({𝑥 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ 𝜑} = {𝐴}))
3	2	spcegv 3554	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∣ 𝜑} = {𝐴} → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}))
4	3	imp 412	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ 𝜑} = {𝐴}) → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
5		euabsn2 4689	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
6	4, 5	sylibr 237	1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ 𝜑} = {𝐴}) → ∃!𝑥𝜑)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃!weu 2595  {cab 2740  {csn 4587

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-sn 4588

This theorem is used by:  rabsneu  4693