Theorem reusn 4695

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 4693	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦{𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦})
2		df-reu 3377	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		df-rab 3424	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
4	3	eqeq1i 2774	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦})
5	4	exbii 1875	. 2 ⊢ (∃𝑦{𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦} ↔ ∃𝑦{𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦})
6	1, 2, 5	3bitr4i 306	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦{𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦})

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∃!weu 2602  {cab 2747  ∃!wreu 3374  {crab 3423  {csn 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-reu 3377  df-rab 3424  df-sn 4592

This theorem is referenced by:  reuen1  9019  cshwrepswhash1  17158  frcond3  30557  vdgn1frgrv2  30584  ddemeas  34567  fineqvnttrclselem1  35453  fineqvnttrclselem2  35454  wevgblacfn  35490