Theorem dfeu2 4334

Description: Alternate definition of existential uniqueness in terms of abstraction. (Contributed by SF, 29-Jan-2015.)

Assertion

Ref	Expression
dfeu2	⊢ (∃!xφ ↔ {x ∣ φ} ∈ 1c)

Proof of Theorem dfeu2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		abbi 2464	. . . 4 ⊢ (∀x(φ ↔ x = y) ↔ {x ∣ φ} = {x ∣ x = y})
2		df-sn 3742	. . . . 5 ⊢ {y} = {x ∣ x = y}
3	2	eqeq2i 2363	. . . 4 ⊢ ({x ∣ φ} = {y} ↔ {x ∣ φ} = {x ∣ x = y})
4	1, 3	bitr4i 243	. . 3 ⊢ (∀x(φ ↔ x = y) ↔ {x ∣ φ} = {y})
5	4	exbii 1582	. 2 ⊢ (∃y∀x(φ ↔ x = y) ↔ ∃y{x ∣ φ} = {y})
6		df-eu 2208	. 2 ⊢ (∃!xφ ↔ ∃y∀x(φ ↔ x = y))
7		el1c 4140	. 2 ⊢ ({x ∣ φ} ∈ 1c ↔ ∃y{x ∣ φ} = {y})
8	5, 6, 7	3bitr4i 268	1 ⊢ (∃!xφ ↔ {x ∣ φ} ∈ 1c)

Syntax hints:   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339  {csn 3738  1_cc1c 4135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-1c 4137

This theorem is referenced by:  euabex  4335  dfiota4  4373