Theorem euabex 5444

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 2608	. 2 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)
2		moabex 5441	. 2 ⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)
3	1, 2	syl 18	1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  ∃*wmo 2567  ∃!weu 2598  {cab 2743  Vcvv 3457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-un 3911  df-in 3913  df-ss 3923  df-sn 4592  df-pr 4594

This theorem is used by:  fineqvnttrclse  35596  tfsconcatun  44124  sprval  48288  prprval  48323