Theorem nfeu1 2583

Description: Bound-variable hypothesis builder for uniqueness. See also nfeu1ALT 2584. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Assertion

Ref	Expression
nfeu1	⊢ Ⅎ𝑥∃!𝑥𝜑

Proof of Theorem nfeu1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eu6 2569	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2		nfa1 2149	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
3	2	nfex 2318	. 2 ⊢ Ⅎ𝑥∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
4	1, 3	nfxfr 1856	1 ⊢ Ⅎ𝑥∃!𝑥𝜑

Syntax hints:   ↔ wb 205  ∀wal 1540  ∃wex 1782  Ⅎwnf 1786  ∃!weu 2563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-11 2155  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-mo 2535  df-eu 2564

This theorem is referenced by:  eupicka  2631  2eu8  2655  nfreu1  3409  eusv2i  5393  eusv2nf  5394  reusv2lem3  5399  iota2  6533  sniota  6535  fv3  6910  tz6.12cOLD  6919  eusvobj1  7402  opiota  8045  dfac5lem5  10122  bnj1366  33840  bnj849  33936  pm14.24  43191  eu2ndop1stv  45833  tz6.12c-afv2  45950  setrec2lem2  47739