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Theorem euabsn 3588
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 (∃!𝑥𝜑 ↔ ∃𝑥{𝑥𝜑} = {𝑥})

Proof of Theorem euabsn
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 euabsn2 3587 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
2 nfv 1508 . . 3 𝑦{𝑥𝜑} = {𝑥}
3 nfab1 2281 . . . 4 𝑥{𝑥𝜑}
43nfeq1 2289 . . 3 𝑥{𝑥𝜑} = {𝑦}
5 sneq 3533 . . . 4 (𝑥 = 𝑦 → {𝑥} = {𝑦})
65eqeq2d 2149 . . 3 (𝑥 = 𝑦 → ({𝑥𝜑} = {𝑥} ↔ {𝑥𝜑} = {𝑦}))
72, 4, 6cbvex 1729 . 2 (∃𝑥{𝑥𝜑} = {𝑥} ↔ ∃𝑦{𝑥𝜑} = {𝑦})
81, 7bitr4i 186 1 (∃!𝑥𝜑 ↔ ∃𝑥{𝑥𝜑} = {𝑥})
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1331  wex 1468  ∃!weu 1997  {cab 2123  {csn 3522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sn 3528
This theorem is referenced by:  eusn  3592  args  4903  mapsn  6577
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