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Theorem euabsn 4532
 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 4531 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
2 nfv 1873 . . 3 𝑦{𝑥𝜑} = {𝑥}
3 nfab1 2928 . . . 4 𝑥{𝑥𝜑}
43nfeq1 2939 . . 3 𝑥{𝑥𝜑} = {𝑦}
5 sneq 4445 . . . 4 (𝑥 = 𝑦 → {𝑥} = {𝑦})
65eqeq2d 2782 . . 3 (𝑥 = 𝑦 → ({𝑥𝜑} = {𝑥} ↔ {𝑥𝜑} = {𝑦}))
72, 4, 6cbvexv1 2278 . 2 (∃𝑥{𝑥𝜑} = {𝑥} ↔ ∃𝑦{𝑥𝜑} = {𝑦})
81, 7bitr4i 270 1 (∃!𝑥𝜑 ↔ ∃𝑥{𝑥𝜑} = {𝑥})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   = wceq 1507  ∃wex 1742  ∃!weu 2583  {cab 2752  {csn 4435 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-sn 4436 This theorem is referenced by:  eusn  4536  uniintsn  4782  args  5794  opabiotadm  6571  mapsnd  8246
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