Theorem euabex 5447

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 2609	. 2 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)
2		moabex 5444	. 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 2568  ∃!weu 2599  {cab 2744  Vcvv 3458

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 2738  ax-sep 5262  ax-pr 5409

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-un 3913  df-in 3915  df-ss 3925  df-sn 4595  df-pr 4597

This theorem is used by:  fineqvnttrclse  35553  tfsconcatun  44097  sprval  48261  prprval  48296