Theorem euabsneu 47765

Description: Another way to express existential uniqueness of a wff 𝜑: its associated class abstraction {𝑥 ∣ 𝜑} is a singleton. Variant of euabsn2 4691 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 4807	. . . 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 538	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ∧ ∃*𝑦{𝑥 ∣ 𝜑} = {𝑦}))
6		euabsn2 4691	. 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})

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809  ∃*wmo 2565  ∃!weu 2596  {cab 2741  {csn 4589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-sn 4590

This theorem is referenced by:  reuaiotaiota  47825