Theorem euabex 4929

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 2499	. 2 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)
2		moabex 4927	. 2 ⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)
3	1, 2	syl 17	1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1990  ∃!weu 2470  ∃*wmo 2471  {cab 2608  Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by:  sprval  41729