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Theorem euabsneu 42097
Description: Another way to express existential uniqueness of a wff 𝜑: its associated class abstraction {𝑥𝜑} is a singleton. Variant of euabsn2 4492 using existential uniqueness for the singleton element instead of existence only. (Contributed by AV, 24-Aug-2022.)
Assertion
Ref Expression
euabsneu (∃!𝑥𝜑 ↔ ∃!𝑦{𝑥𝜑} = {𝑦})
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem euabsneu
StepHypRef Expression
1 mosneq 4602 . . . 4 ∃*𝑦{𝑦} = {𝑥𝜑}
2 eqcom 2785 . . . . 5 ({𝑦} = {𝑥𝜑} ↔ {𝑥𝜑} = {𝑦})
32mobii 2563 . . . 4 (∃*𝑦{𝑦} = {𝑥𝜑} ↔ ∃*𝑦{𝑥𝜑} = {𝑦})
41, 3mpbi 222 . . 3 ∃*𝑦{𝑥𝜑} = {𝑦}
54biantru 525 . 2 (∃𝑦{𝑥𝜑} = {𝑦} ↔ (∃𝑦{𝑥𝜑} = {𝑦} ∧ ∃*𝑦{𝑥𝜑} = {𝑦}))
6 euabsn2 4492 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
7 df-eu 2587 . 2 (∃!𝑦{𝑥𝜑} = {𝑦} ↔ (∃𝑦{𝑥𝜑} = {𝑦} ∧ ∃*𝑦{𝑥𝜑} = {𝑦}))
85, 6, 73bitr4i 295 1 (∃!𝑥𝜑 ↔ ∃!𝑦{𝑥𝜑} = {𝑦})
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386   = wceq 1601  wex 1823  ∃*wmo 2549  ∃!weu 2586  {cab 2763  {csn 4398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-v 3400  df-sn 4399
This theorem is referenced by:  reuaiotaiota  42118
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