Theorem euabex 4254

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 3687	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
2		vex 2763	. . . . 5 ⊢ 𝑦 ∈ V
3	2	snex 4214	. . . 4 ⊢ {𝑦} ∈ V
4		eleq1 2256	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} ∈ V ↔ {𝑦} ∈ V))
5	3, 4	mpbiri 168	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → {𝑥 ∣ 𝜑} ∈ V)
6	5	exlimiv 1609	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → {𝑥 ∣ 𝜑} ∈ V)
7	1, 6	sylbi 121	1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4   = wceq 1364  ∃wex 1503  ∃!weu 2042   ∈ wcel 2164  {cab 2179  Vcvv 2760  {csn 3618

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624

This theorem is referenced by: (None)