Theorem euabsneu 43620

Description: Another way to express existential uniqueness of a wff 𝜑: its associated class abstraction {𝑥 ∣ 𝜑} is a singleton. Variant of euabsn2 4621 using existential uniqueness for the singleton element instead of existence only. (Contributed by AV, 24-Aug-2022.)

Assertion

Ref	Expression
euabsneu	⊢ (∃!𝑥𝜑 ↔ ∃!𝑦{𝑥 ∣ 𝜑} = {𝑦})

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem euabsneu

Step	Hyp	Ref	Expression
1		mosneq 4733	. . . 4 ⊢ ∃*𝑦{𝑦} = {𝑥 ∣ 𝜑}
2		eqcom 2805	. . . . 5 ⊢ ({𝑦} = {𝑥 ∣ 𝜑} ↔ {𝑥 ∣ 𝜑} = {𝑦})
3	2	mobii 2606	. . . 4 ⊢ (∃*𝑦{𝑦} = {𝑥 ∣ 𝜑} ↔ ∃*𝑦{𝑥 ∣ 𝜑} = {𝑦})
4	1, 3	mpbi 233	. . 3 ⊢ ∃*𝑦{𝑥 ∣ 𝜑} = {𝑦}
5	4	biantru 533	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ∧ ∃*𝑦{𝑥 ∣ 𝜑} = {𝑦}))
6		euabsn2 4621	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
7		df-eu 2629	. 2 ⊢ (∃!𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ∧ ∃*𝑦{𝑥 ∣ 𝜑} = {𝑦}))
8	5, 6, 7	3bitr4i 306	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦{𝑥 ∣ 𝜑} = {𝑦})

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781  ∃*wmo 2596  ∃!weu 2628  {cab 2776  {csn 4525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-sn 4526

This theorem is referenced by:  reuaiotaiota  43645