Theorem nfeu1 2214

Description: Bound-variable hypothesis builder for uniqueness. (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		df-eu 2208	. 2 ⊢ (∃!xφ ↔ ∃y∀x(φ ↔ x = y))
2		nfa1 1788	. . 3 ⊢ Ⅎx∀x(φ ↔ x = y)
3	2	nfex 1843	. 2 ⊢ Ⅎx∃y∀x(φ ↔ x = y)
4	1, 3	nfxfr 1570	1 ⊢ Ⅎx∃!xφ

Syntax hints:   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  ∃!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545  df-eu 2208

This theorem is referenced by:  nfmo1  2215  moaneu  2263  eupicka  2268  2eu8  2291  exists2  2294  nfreu1  2782  iota2  4368  sniota  4370  fv3  5342  tz6.12c  5348