Theorem euabsneu 48371

Description: Another way to express existential uniqueness of a wff 𝜑: its associated class abstraction {𝑥 ∣ 𝜑} is a singleton. Variant of euabsn2 4688 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 4802	. . . 4 ⊢ ∃*𝑦{𝑦} = {𝑥 ∣ 𝜑}
2		eqcom 2770	. . . . 5 ⊢ ({𝑦} = {𝑥 ∣ 𝜑} ↔ {𝑥 ∣ 𝜑} = {𝑦})
3	2	mobii 2576	. . . 4 ⊢ (∃*𝑦{𝑦} = {𝑥 ∣ 𝜑} ↔ ∃*𝑦{𝑥 ∣ 𝜑} = {𝑦})
4	1, 3	mpbi 233	. . 3 ⊢ ∃*𝑦{𝑥 ∣ 𝜑} = {𝑦}
5	4	biantru 539	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ∧ ∃*𝑦{𝑥 ∣ 𝜑} = {𝑦}))
6		euabsn2 4688	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
7		df-eu 2597	. 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})

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812  ∃*wmo 2565  ∃!weu 2596  {cab 2741  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-sn 4589

This theorem is used by:  reuaiotaiota  48431