Theorem euabex 4323

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 3744	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
2		vex 2806	. . . . 5 ⊢ 𝑦 ∈ V
3	2	snex 4281	. . . 4 ⊢ {𝑦} ∈ V
4		eleq1 2294	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} ∈ V ↔ {𝑦} ∈ V))
5	3, 4	mpbiri 168	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → {𝑥 ∣ 𝜑} ∈ V)
6	5	exlimiv 1647	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → {𝑥 ∣ 𝜑} ∈ V)
7	1, 6	sylbi 121	1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4   = wceq 1398  ∃wex 1541  ∃!weu 2079   ∈ wcel 2202  {cab 2217  Vcvv 2803  {csn 3673

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679

This theorem is referenced by: (None)