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Theorem euabsn 3792
 Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)
Assertion
Ref Expression
euabsn (∃!xφx{x φ} = {x})

Proof of Theorem euabsn
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 euabsn2 3791 . 2 (∃!xφy{x φ} = {y})
2 nfv 1619 . . 3 y{x φ} = {x}
3 nfab1 2491 . . . 4 x{x φ}
43nfeq1 2498 . . 3 x{x φ} = {y}
5 sneq 3744 . . . 4 (x = y → {x} = {y})
65eqeq2d 2364 . . 3 (x = y → ({x φ} = {x} ↔ {x φ} = {y}))
72, 4, 6cbvex 1985 . 2 (x{x φ} = {x} ↔ y{x φ} = {y})
81, 7bitr4i 243 1 (∃!xφx{x φ} = {x})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∃wex 1541   = wceq 1642  ∃!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-or 359  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-nfc 2478  df-sn 3741 This theorem is referenced by:  eusn  3796  uniintsn  3963  mapsn  6026
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