Theorem euabex 3913

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 3390	. 2 ⊢ (∃!xφ ↔ ∃y{x ∣ φ} = {y})
2		vex 2536	. . . . 5 ⊢ y ∈ V
3		snexgOLD 3887	. . . . 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 1472	. 2 ⊢ (∃y{x ∣ φ} = {y} → {x ∣ φ} ∈ V)
8	1, 7	sylbi 114	1 ⊢ (∃!xφ → {x ∣ φ} ∈ V)

Syntax hints:   → wi 4  ∃wex 1361   = wceq 1373   ∈ wcel 1375  ∃!weu 1881  {cab 2008  Vcvv 2533  {csn 3327

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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-sep 3827  ax-pow 3879

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1329  df-sb 1628  df-eu 1884  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2535  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333

This theorem is referenced by: (None)