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Theorem nfeud2OLD 2623
 Description: Obsolete proof of nfeud2 2611 as of 14-Oct-2022. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfeud2OLD.1 𝑦𝜑
nfeud2OLD.2 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfeud2OLD (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Proof of Theorem nfeud2OLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eu6 2592 . 2 (∃!𝑦𝜓 ↔ ∃𝑧𝑦(𝜓𝑦 = 𝑧))
2 nfv 1957 . . 3 𝑧𝜑
3 nfeud2OLD.1 . . . 4 𝑦𝜑
4 nfeud2OLD.2 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
5 nfeqf1 2343 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
65adantl 475 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 = 𝑧)
74, 6nfbid 1949 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝜓𝑦 = 𝑧))
83, 7nfald2 2411 . . 3 (𝜑 → Ⅎ𝑥𝑦(𝜓𝑦 = 𝑧))
92, 8nfexd 2305 . 2 (𝜑 → Ⅎ𝑥𝑧𝑦(𝜓𝑦 = 𝑧))
101, 9nfxfrd 1898 1 (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1599  ∃wex 1823  Ⅎwnf 1827  ∃!weu 2586 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-mo 2551  df-eu 2587 This theorem is referenced by: (None)
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