Theorem euabsn2 4693

Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
euabsn2	⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem euabsn2

Step	Hyp	Ref	Expression
1		eu6 2608	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2		absn 4611	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
3	2	exbii 1875	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
4	1, 3	bitr4i 281	1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})

Syntax hints:   ↔ wb 209  ∀wal 1565   = wceq 1567  ∃wex 1806  ∃!weu 2602  {cab 2747  {csn 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-sn 4592

This theorem is referenced by:  euabsn  4694  reusn  4695  absneu  4696  uniintab  4952  eusvobj2  7400  euabsneu  47647  aiotaexb  47708