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Theorem euabsn 4403
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 4402 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
2 nfv 1990 . . 3 𝑦{𝑥𝜑} = {𝑥}
3 nfab1 2902 . . . 4 𝑥{𝑥𝜑}
43nfeq1 2914 . . 3 𝑥{𝑥𝜑} = {𝑦}
5 sneq 4329 . . . 4 (𝑥 = 𝑦 → {𝑥} = {𝑦})
65eqeq2d 2768 . . 3 (𝑥 = 𝑦 → ({𝑥𝜑} = {𝑥} ↔ {𝑥𝜑} = {𝑦}))
72, 4, 6cbvex 2415 . 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 1630  wex 1851  ∃!weu 2605  {cab 2744  {csn 4319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-v 3340  df-sn 4320
This theorem is referenced by:  eusn  4407  uniintsn  4664  args  5649  opabiotadm  6420  mapsn  8063  mapsnd  39885
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