New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  dfeu2 GIF version

Theorem dfeu2 4333
 Description: Alternate definition of existential uniqueness in terms of abstraction. (Contributed by SF, 29-Jan-2015.)
Assertion
Ref Expression
dfeu2 (∃!xφ ↔ {x φ} 1c)

Proof of Theorem dfeu2
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 abbi 2463 . . . 4 (x(φx = y) ↔ {x φ} = {x x = y})
2 df-sn 3741 . . . . 5 {y} = {x x = y}
32eqeq2i 2363 . . . 4 ({x φ} = {y} ↔ {x φ} = {x x = y})
41, 3bitr4i 243 . . 3 (x(φx = y) ↔ {x φ} = {y})
54exbii 1582 . 2 (yx(φx = y) ↔ y{x φ} = {y})
6 df-eu 2208 . 2 (∃!xφyx(φx = y))
7 el1c 4139 . 2 ({x φ} 1cy{x φ} = {y})
85, 6, 73bitr4i 268 1 (∃!xφ ↔ {x φ} 1c)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339  {csn 3737  1cc1c 4134 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  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  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-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-1c 4136 This theorem is referenced by:  euabex  4334  dfiota4  4372
 Copyright terms: Public domain W3C validator