Theorem absneu 3779

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 3716	. . . . 5 ⊢ (𝑦 = 𝐴 → {𝑦} = {𝐴})
2	1	eqeq2d 2250	. . . 4 ⊢ (𝑦 = 𝐴 → ({𝑥 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ 𝜑} = {𝐴}))
3	2	spcegv 2913	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∣ 𝜑} = {𝐴} → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}))
4	3	imp 124	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ 𝜑} = {𝐴}) → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
5		euabsn2 3776	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
6	4, 5	sylibr 134	1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ 𝜑} = {𝐴}) → ∃!𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402  ∃wex 1545  ∃!weu 2086   ∈ wcel 2209  {cab 2224  {csn 3705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sn 3711

This theorem is referenced by:  rabsneu  3780