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Theorem wl-eutf 33326
Description: Closed form of df-eu 2472 with a distinctor avoiding distinct variable conditions. (Contributed by Wolf Lammen, 23-Sep-2020.)
Assertion
Ref Expression
wl-eutf ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))

Proof of Theorem wl-eutf
StepHypRef Expression
1 nfnae 2316 . . 3 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2 nfa1 2026 . . 3 𝑥𝑥𝑦𝜑
31, 2nfan 1826 . 2 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑)
4 nfnae 2316 . . 3 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
5 nfnf1 2029 . . . 4 𝑦𝑦𝜑
65nfal 2151 . . 3 𝑦𝑥𝑦𝜑
74, 6nfan 1826 . 2 𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑)
8 simpl 473 . 2 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → ¬ ∀𝑥 𝑥 = 𝑦)
9 sp 2051 . . 3 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝜑)
109adantl 482 . 2 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → Ⅎ𝑦𝜑)
113, 7, 8, 10wl-eudf 33325 1 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1479  wex 1702  wnf 1706  ∃!weu 2468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-eu 2472
This theorem is referenced by: (None)
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