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Theorem euabsn 3465
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 3464 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
2 nfv 1435 . . 3 𝑦{𝑥𝜑} = {𝑥}
3 nfab1 2194 . . . 4 𝑥{𝑥𝜑}
43nfeq1 2201 . . 3 𝑥{𝑥𝜑} = {𝑦}
5 sneq 3411 . . . 4 (𝑥 = 𝑦 → {𝑥} = {𝑦})
65eqeq2d 2065 . . 3 (𝑥 = 𝑦 → ({𝑥𝜑} = {𝑥} ↔ {𝑥𝜑} = {𝑦}))
72, 4, 6cbvex 1653 . 2 (∃𝑥{𝑥𝜑} = {𝑥} ↔ ∃𝑦{𝑥𝜑} = {𝑦})
81, 7bitr4i 180 1 (∃!𝑥𝜑 ↔ ∃𝑥{𝑥𝜑} = {𝑥})
Colors of variables: wff set class
Syntax hints:  wb 102   = wceq 1257  wex 1395  ∃!weu 1914  {cab 2040  {csn 3400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-sn 3406
This theorem is referenced by:  eusn  3469  args  4719
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