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Theorem euabsn2 3791
 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 (∃!xφy{x φ} = {y})
Distinct variable groups:   x,y   φ,y
Allowed substitution hint:   φ(x)

Proof of Theorem euabsn2
StepHypRef Expression
1 df-eu 2208 . 2 (∃!xφyx(φx = y))
2 abeq1 2459 . . . 4 ({x φ} = {y} ↔ x(φx {y}))
3 elsn 3748 . . . . . 6 (x {y} ↔ x = y)
43bibi2i 304 . . . . 5 ((φx {y}) ↔ (φx = y))
54albii 1566 . . . 4 (x(φx {y}) ↔ x(φx = y))
62, 5bitri 240 . . 3 ({x φ} = {y} ↔ x(φx = y))
76exbii 1582 . 2 (y{x φ} = {y} ↔ yx(φx = y))
81, 7bitr4i 243 1 (∃!xφy{x φ} = {y})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 3741 This theorem is referenced by:  euabsn  3792  reusn  3793  absneu  3794  uniintab  3964
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