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Theorem euabsn 4231
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 4230 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
2 nfv 1840 . . 3 𝑦{𝑥𝜑} = {𝑥}
3 nfab1 2763 . . . 4 𝑥{𝑥𝜑}
43nfeq1 2774 . . 3 𝑥{𝑥𝜑} = {𝑦}
5 sneq 4158 . . . 4 (𝑥 = 𝑦 → {𝑥} = {𝑦})
65eqeq2d 2631 . . 3 (𝑥 = 𝑦 → ({𝑥𝜑} = {𝑥} ↔ {𝑥𝜑} = {𝑦}))
72, 4, 6cbvex 2271 . 2 (∃𝑥{𝑥𝜑} = {𝑥} ↔ ∃𝑦{𝑥𝜑} = {𝑦})
81, 7bitr4i 267 1 (∃!𝑥𝜑 ↔ ∃𝑥{𝑥𝜑} = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wex 1701  ∃!weu 2469  {cab 2607  {csn 4148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-sn 4149
This theorem is referenced by:  eusn  4235  uniintsn  4479  args  5452  opabiotadm  6217  mapsn  7843  mapsnd  38862
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