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Theorem euabsn 4647
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 4646 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
2 nfv 1916 . . 3 𝑦{𝑥𝜑} = {𝑥}
3 nfab1 2984 . . . 4 𝑥{𝑥𝜑}
43nfeq1 2997 . . 3 𝑥{𝑥𝜑} = {𝑦}
5 sneq 4560 . . . 4 (𝑥 = 𝑦 → {𝑥} = {𝑦})
65eqeq2d 2835 . . 3 (𝑥 = 𝑦 → ({𝑥𝜑} = {𝑥} ↔ {𝑥𝜑} = {𝑦}))
72, 4, 6cbvexv1 2364 . 2 (∃𝑥{𝑥𝜑} = {𝑥} ↔ ∃𝑦{𝑥𝜑} = {𝑦})
81, 7bitr4i 281 1 (∃!𝑥𝜑 ↔ ∃𝑥{𝑥𝜑} = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wex 1781  ∃!weu 2654  {cab 2802  {csn 4550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-nfc 2964  df-sn 4551
This theorem is referenced by:  eusn  4651  uniintsn  4899  args  5946  opabiotadm  6738  mapsnd  8448
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