Theorem euabex 4334

Description: If there is a unique object satisfying a property φ, then the set of all elements that satisfy φ exists. (Contributed by SF, 16-Jan-2015.)

Assertion

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

Proof of Theorem euabex

Step	Hyp	Ref	Expression
1		dfeu2 4333	. 2 ⊢ (∃!xφ ↔ {x ∣ φ} ∈ 1c)
2		elex 2867	. 2 ⊢ ({x ∣ φ} ∈ 1c → {x ∣ φ} ∈ V)
3	1, 2	sylbi 187	1 ⊢ (∃!xφ → {x ∣ φ} ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1710  ∃!weu 2204  {cab 2339  Vcvv 2859  1_cc1c 4134

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 4078  ax-sn 4087

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 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-1c 4136

This theorem is referenced by:  dfiota4  4372