Theorem euabex 4268

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 3701	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
2		vex 2774	. . . . 5 ⊢ 𝑦 ∈ V
3	2	snex 4228	. . . 4 ⊢ {𝑦} ∈ V
4		eleq1 2267	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} ∈ V ↔ {𝑦} ∈ V))
5	3, 4	mpbiri 168	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → {𝑥 ∣ 𝜑} ∈ V)
6	5	exlimiv 1620	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → {𝑥 ∣ 𝜑} ∈ V)
7	1, 6	sylbi 121	1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4   = wceq 1372  ∃wex 1514  ∃!weu 2053   ∈ wcel 2175  {cab 2190  Vcvv 2771  {csn 3632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638

This theorem is referenced by: (None)