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Theorem euabsn 4692
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 4691 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
2 nfv 1918 . . 3 𝑦{𝑥𝜑} = {𝑥}
3 nfab1 2910 . . . 4 𝑥{𝑥𝜑}
43nfeq1 2923 . . 3 𝑥{𝑥𝜑} = {𝑦}
5 sneq 4601 . . . 4 (𝑥 = 𝑦 → {𝑥} = {𝑦})
65eqeq2d 2748 . . 3 (𝑥 = 𝑦 → ({𝑥𝜑} = {𝑥} ↔ {𝑥𝜑} = {𝑦}))
72, 4, 6cbvexv1 2339 . 2 (∃𝑥{𝑥𝜑} = {𝑥} ↔ ∃𝑦{𝑥𝜑} = {𝑦})
81, 7bitr4i 278 1 (∃!𝑥𝜑 ↔ ∃𝑥{𝑥𝜑} = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wex 1782  ∃!weu 2567  {cab 2714  {csn 4591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-nfc 2890  df-sn 4592
This theorem is referenced by:  eusn  4696  uniintsn  4953  args  6049  opabiotadm  6928  mapsnd  8831
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