Theorem euabex 5481

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 2581	. 2 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)
2		moabex 5479	. 2 ⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)
3	1, 2	syl 17	1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2108  ∃*wmo 2541  ∃!weu 2571  {cab 2717  Vcvv 3488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-rab 3444  df-v 3490  df-un 3981  df-in 3983  df-ss 3993  df-sn 4649  df-pr 4651

This theorem is referenced by:  tfsconcatun  43299  sprval  47353  prprval  47388