Theorem euabsn2 3531

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		df-eu 1958	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2		abeq1 2204	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ {𝑦}))
3		velsn 3483	. . . . . 6 ⊢ (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦)
4	3	bibi2i 226	. . . . 5 ⊢ ((𝜑 ↔ 𝑥 ∈ {𝑦}) ↔ (𝜑 ↔ 𝑥 = 𝑦))
5	4	albii 1411	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 ∈ {𝑦}) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
6	2, 5	bitri 183	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
7	6	exbii 1548	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
8	1, 7	bitr4i 186	1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})

Syntax hints:   ↔ wb 104  ∀wal 1294   = wceq 1296  ∃wex 1433   ∈ wcel 1445  ∃!weu 1955  {cab 2081  {csn 3466

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-sn 3472

This theorem is referenced by:  euabsn  3532  reusn  3533  absneu  3534  uniintabim  3747  euabex  4076  nfvres  5372  eusvobj2  5676