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Theorem euabsneu 43262
 Description: Another way to express existential uniqueness of a wff 𝜑: its associated class abstraction {𝑥 ∣ 𝜑} is a singleton. Variant of euabsn2 4660 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 4772 . . . 4 ∃*𝑦{𝑦} = {𝑥𝜑}
2 eqcom 2828 . . . . 5 ({𝑦} = {𝑥𝜑} ↔ {𝑥𝜑} = {𝑦})
32mobii 2627 . . . 4 (∃*𝑦{𝑦} = {𝑥𝜑} ↔ ∃*𝑦{𝑥𝜑} = {𝑦})
41, 3mpbi 232 . . 3 ∃*𝑦{𝑥𝜑} = {𝑦}
54biantru 532 . 2 (∃𝑦{𝑥𝜑} = {𝑦} ↔ (∃𝑦{𝑥𝜑} = {𝑦} ∧ ∃*𝑦{𝑥𝜑} = {𝑦}))
6 euabsn2 4660 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
7 df-eu 2650 . 2 (∃!𝑦{𝑥𝜑} = {𝑦} ↔ (∃𝑦{𝑥𝜑} = {𝑦} ∧ ∃*𝑦{𝑥𝜑} = {𝑦}))
85, 6, 73bitr4i 305 1 (∃!𝑥𝜑 ↔ ∃!𝑦{𝑥𝜑} = {𝑦})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776  ∃*wmo 2616  ∃!weu 2649  {cab 2799  {csn 4566 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-sn 4567 This theorem is referenced by:  reuaiotaiota  43287
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