Theorem nfeud2 2216

Description: Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.)

Hypotheses

Ref	Expression
nfeud2.1	⊢ Ⅎyφ
nfeud2.2	⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎxψ)

Assertion

Ref	Expression
nfeud2	⊢ (φ → Ⅎx∃!yψ)

Proof of Theorem nfeud2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2208	. 2 ⊢ (∃!yψ ↔ ∃z∀y(ψ ↔ y = z))
2		nfv 1619	. . 3 ⊢ Ⅎzφ
3		nfeud2.1	. . . . 5 ⊢ Ⅎyφ
4		nfnae 1956	. . . . 5 ⊢ Ⅎy ¬ ∀x x = z
5	3, 4	nfan 1824	. . . 4 ⊢ Ⅎy(φ ∧ ¬ ∀x x = z)
6		nfeud2.2	. . . . . 6 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎxψ)
7	6	adantlr 695	. . . . 5 ⊢ (((φ ∧ ¬ ∀x x = z) ∧ ¬ ∀x x = y) → Ⅎxψ)
8		nfeqf 1958	. . . . . . 7 ⊢ ((¬ ∀x x = y ∧ ¬ ∀x x = z) → Ⅎx y = z)
9	8	ancoms 439	. . . . . 6 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = y) → Ⅎx y = z)
10	9	adantll 694	. . . . 5 ⊢ (((φ ∧ ¬ ∀x x = z) ∧ ¬ ∀x x = y) → Ⅎx y = z)
11	7, 10	nfbid 1832	. . . 4 ⊢ (((φ ∧ ¬ ∀x x = z) ∧ ¬ ∀x x = y) → Ⅎx(ψ ↔ y = z))
12	5, 11	nfald2 1972	. . 3 ⊢ ((φ ∧ ¬ ∀x x = z) → Ⅎx∀y(ψ ↔ y = z))
13	2, 12	nfexd2 1973	. 2 ⊢ (φ → Ⅎx∃z∀y(ψ ↔ y = z))
14	1, 13	nfxfrd 1571	1 ⊢ (φ → Ⅎx∃!yψ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀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  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-eu 2208

This theorem is referenced by:  nfmod2  2217  nfeud  2218  nfreud  2784