Theorem reusn 4708

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 4706	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦{𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦})
2		df-reu 3365	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		df-rab 3421	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
4	3	eqeq1i 2741	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦})
5	4	exbii 1848	. 2 ⊢ (∃𝑦{𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦} ↔ ∃𝑦{𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦})
6	1, 2, 5	3bitr4i 303	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦{𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦})

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃!weu 2568  {cab 2714  ∃!wreu 3362  {crab 3420  {csn 4606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-reu 3365  df-rab 3421  df-sn 4607

This theorem is referenced by:  reuen1  9045  cshwrepswhash1  17127  frcond3  30255  vdgn1frgrv2  30282  ddemeas  34272  wevgblacfn  35136