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Theorem nfeud2 2675
 Description: Bound-variable hypothesis builder for uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2390. Check out nfeudw 2676 for a version that replaces the distinctor with a disjoint variable condition, not requiring ax-13 2390. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.) (Proof shortened by BJ, 14-Oct-2022.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfeud2.1 𝑦𝜑
nfeud2.2 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfeud2 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Proof of Theorem nfeud2
StepHypRef Expression
1 df-eu 2653 . 2 (∃!𝑦𝜓 ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓))
2 nfeud2.1 . . . 4 𝑦𝜑
3 nfeud2.2 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
42, 3nfexd2 2468 . . 3 (𝜑 → Ⅎ𝑥𝑦𝜓)
52, 3nfmod2 2641 . . 3 (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
64, 5nfand 1898 . 2 (𝜑 → Ⅎ𝑥(∃𝑦𝜓 ∧ ∃*𝑦𝜓))
71, 6nfxfrd 1854 1 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398  ∀wal 1535  ∃wex 1780  Ⅎwnf 1784  ∃*wmo 2620  ∃!weu 2652 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-mo 2622  df-eu 2653 This theorem is referenced by:  nfeud  2677  nfreud  3359
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