Theorem euabex 5407

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

Step	Hyp	Ref	Expression
1		eumo 2576	. 2 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)
2		moabex 5404	. 2 ⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)
3	1, 2	syl 17	1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2113  ∃*wmo 2535  ∃!weu 2566  {cab 2712  Vcvv 3438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-un 3904  df-in 3906  df-ss 3916  df-sn 4579  df-pr 4581

This theorem is referenced by:  fineqvnttrclse  35229  tfsconcatun  43521  sprval  47667  prprval  47702