Theorem reusn 3793

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 3791	. 2 ⊢ (∃!x(x ∈ A ∧ φ) ↔ ∃y{x ∣ (x ∈ A ∧ φ)} = {y})
2		df-reu 2621	. 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
3		df-rab 2623	. . . 4 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
4	3	eqeq1i 2360	. . 3 ⊢ ({x ∈ A ∣ φ} = {y} ↔ {x ∣ (x ∈ A ∧ φ)} = {y})
5	4	exbii 1582	. 2 ⊢ (∃y{x ∈ A ∣ φ} = {y} ↔ ∃y{x ∣ (x ∈ A ∧ φ)} = {y})
6	1, 2, 5	3bitr4i 268	1 ⊢ (∃!x ∈ A φ ↔ ∃y{x ∈ A ∣ φ} = {y})

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339  ∃!wreu 2616  {crab 2618  {csn 3737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-reu 2621  df-rab 2623  df-sn 3741

This theorem is referenced by: (None)