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Theorem nfeud2 2216
 Description: Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
nfeud2.1 yφ
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.
StepHypRef Expression
1 df-eu 2208 . 2 (∃!yψzy(ψy = z))
2 nfv 1619 . . 3 zφ
3 nfeud2.1 . . . . 5 yφ
4 nfnae 1956 . . . . 5 y ¬ x x = z
53, 4nfan 1824 . . . 4 y(φ ¬ x x = z)
6 nfeud2.2 . . . . . 6 ((φ ¬ x x = y) → Ⅎxψ)
76adantlr 695 . . . . 5 (((φ ¬ x x = z) ¬ x x = y) → Ⅎxψ)
8 nfeqf 1958 . . . . . . 7 ((¬ x x = y ¬ x x = z) → Ⅎx y = z)
98ancoms 439 . . . . . 6 ((¬ x x = z ¬ x x = y) → Ⅎx y = z)
109adantll 694 . . . . 5 (((φ ¬ x x = z) ¬ x x = y) → Ⅎx y = z)
117, 10nfbid 1832 . . . 4 (((φ ¬ x x = z) ¬ x x = y) → Ⅎx(ψy = z))
125, 11nfald2 1972 . . 3 ((φ ¬ x x = z) → Ⅎxy(ψy = z))
132, 12nfexd2 1973 . 2 (φ → Ⅎxzy(ψy = z))
141, 13nfxfrd 1571 1 (φ → Ⅎx∃!yψ)
 Colors of variables: wff setvar class 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-1 5  ax-2 6  ax-3 7  ax-mp 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  2783
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