Theorem reusn 3678

Description: A way to express restricted existential uniqueness of a wff: its restricted class abstraction is a singleton. (Contributed by NM, 30-May-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
reusn	⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦{𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦})

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem reusn

Step	Hyp	Ref	Expression
1		euabsn2 3676	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦{𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦})
2		df-reu 2475	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		df-rab 2477	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
4	3	eqeq1i 2197	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦})
5	4	exbii 1616	. 2 ⊢ (∃𝑦{𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦} ↔ ∃𝑦{𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦})
6	1, 2, 5	3bitr4i 212	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦{𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦})

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1503  ∃!weu 2038   ∈ wcel 2160  {cab 2175  ∃!wreu 2470  {crab 2472  {csn 3607

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-reu 2475  df-rab 2477  df-v 2754  df-sn 3613

This theorem is referenced by:  reuen1  6822