Theorem euabex 4277

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 3707	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
2		vex 2776	. . . . 5 ⊢ 𝑦 ∈ V
3	2	snex 4237	. . . 4 ⊢ {𝑦} ∈ V
4		eleq1 2269	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} ∈ V ↔ {𝑦} ∈ V))
5	3, 4	mpbiri 168	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → {𝑥 ∣ 𝜑} ∈ V)
6	5	exlimiv 1622	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → {𝑥 ∣ 𝜑} ∈ V)
7	1, 6	sylbi 121	1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4   = wceq 1373  ∃wex 1516  ∃!weu 2055   ∈ wcel 2177  {cab 2192  Vcvv 2773  {csn 3638

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644

This theorem is referenced by: (None)