Theorem euabex 3933

Description: The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)

Assertion

Ref	Expression
euabex	⊢ (∃!xφ → {x ∣ φ} ∈ V)

Proof of Theorem euabex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		euabsn2 3411	. 2 ⊢ (∃!xφ ↔ ∃y{x ∣ φ} = {y})
2		vex 2536	. . . . 5 ⊢ y ∈ V
3		snexgOLD 3907	. . . . 5 ⊢ (y ∈ V → {y} ∈ V)
4	2, 3	ax-mp 7	. . . 4 ⊢ {y} ∈ V
5		eleq1 2082	. . . 4 ⊢ ({x ∣ φ} = {y} → ({x ∣ φ} ∈ V ↔ {y} ∈ V))
6	4, 5	mpbiri 157	. . 3 ⊢ ({x ∣ φ} = {y} → {x ∣ φ} ∈ V)
7	6	exlimiv 1471	. 2 ⊢ (∃y{x ∣ φ} = {y} → {x ∣ φ} ∈ V)
8	1, 7	sylbi 114	1 ⊢ (∃!xφ → {x ∣ φ} ∈ V)

Syntax hints:   → wi 4   = wceq 1228  ∃wex 1362   ∈ wcel 1374  ∃!weu 1882  {cab 2008  Vcvv 2533  {csn 3348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3847  ax-pow 3899

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2535  df-in 2899  df-ss 2906  df-pw 3334  df-sn 3354

This theorem is referenced by: (None)