Theorem euabex 3952

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 3430	. 2 ⊢ (∃!xφ ↔ ∃y{x ∣ φ} = {y})
2		vex 2554	. . . . 5 ⊢ y ∈ V
3		snexgOLD 3926	. . . . 5 ⊢ (y ∈ V → {y} ∈ V)
4	2, 3	ax-mp 7	. . . 4 ⊢ {y} ∈ V
5		eleq1 2097	. . . 4 ⊢ ({x ∣ φ} = {y} → ({x ∣ φ} ∈ V ↔ {y} ∈ V))
6	4, 5	mpbiri 157	. . 3 ⊢ ({x ∣ φ} = {y} → {x ∣ φ} ∈ V)
7	6	exlimiv 1486	. 2 ⊢ (∃y{x ∣ φ} = {y} → {x ∣ φ} ∈ V)
8	1, 7	sylbi 114	1 ⊢ (∃!xφ → {x ∣ φ} ∈ V)

Syntax hints:   → wi 4   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃!weu 1897  {cab 2023  Vcvv 2551  {csn 3367

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373

This theorem is referenced by: (None)