Theorem euabsn2 3792

Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
euabsn2	⊢ (∃!xφ ↔ ∃y{x ∣ φ} = {y})

Distinct variable groups:   x,y   φ,y

Allowed substitution hint:   φ(x)

Proof of Theorem euabsn2

Step	Hyp	Ref	Expression
1		df-eu 2208	. 2 ⊢ (∃!xφ ↔ ∃y∀x(φ ↔ x = y))
2		eqabcb 2460	. . . 4 ⊢ ({x ∣ φ} = {y} ↔ ∀x(φ ↔ x ∈ {y}))
3		elsn 3749	. . . . . 6 ⊢ (x ∈ {y} ↔ x = y)
4	3	bibi2i 304	. . . . 5 ⊢ ((φ ↔ x ∈ {y}) ↔ (φ ↔ x = y))
5	4	albii 1566	. . . 4 ⊢ (∀x(φ ↔ x ∈ {y}) ↔ ∀x(φ ↔ x = y))
6	2, 5	bitri 240	. . 3 ⊢ ({x ∣ φ} = {y} ↔ ∀x(φ ↔ x = y))
7	6	exbii 1582	. 2 ⊢ (∃y{x ∣ φ} = {y} ↔ ∃y∀x(φ ↔ x = y))
8	1, 7	bitr4i 243	1 ⊢ (∃!xφ ↔ ∃y{x ∣ φ} = {y})

Syntax hints:   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339  {csn 3738

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-sn 3742

This theorem is referenced by:  euabsn  3793  reusn  3794  absneu  3795  uniintab  3965