Theorem euabsneu 45738

Description: Another way to express existential uniqueness of a wff 𝜑: its associated class abstraction {𝑥 ∣ 𝜑} is a singleton. Variant of euabsn2 4730 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 4844	. . . 4 ⊢ ∃*𝑦{𝑦} = {𝑥 ∣ 𝜑}
2		eqcom 2740	. . . . 5 ⊢ ({𝑦} = {𝑥 ∣ 𝜑} ↔ {𝑥 ∣ 𝜑} = {𝑦})
3	2	mobii 2543	. . . 4 ⊢ (∃*𝑦{𝑦} = {𝑥 ∣ 𝜑} ↔ ∃*𝑦{𝑥 ∣ 𝜑} = {𝑦})
4	1, 3	mpbi 229	. . 3 ⊢ ∃*𝑦{𝑥 ∣ 𝜑} = {𝑦}
5	4	biantru 531	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ∧ ∃*𝑦{𝑥 ∣ 𝜑} = {𝑦}))
6		euabsn2 4730	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
7		df-eu 2564	. 2 ⊢ (∃!𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ∧ ∃*𝑦{𝑥 ∣ 𝜑} = {𝑦}))
8	5, 6, 7	3bitr4i 303	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦{𝑥 ∣ 𝜑} = {𝑦})

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542  ∃wex 1782  ∃*wmo 2533  ∃!weu 2563  {cab 2710  {csn 4629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-sn 4630

This theorem is referenced by:  reuaiotaiota  45796