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Theorem wl-eutf 34874
Description: Closed form of eu6 2660 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 2458 . . 3 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2 nfa1 2156 . . 3 𝑥𝑥𝑦𝜑
31, 2nfan 1901 . 2 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑)
4 nfnae 2458 . . 3 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
5 nfnf1 2159 . . . 4 𝑦𝑦𝜑
65nfal 2344 . . 3 𝑦𝑥𝑦𝜑
74, 6nfan 1901 . 2 𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑)
8 simpl 486 . 2 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → ¬ ∀𝑥 𝑥 = 𝑦)
9 sp 2184 . . 3 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝜑)
109adantl 485 . 2 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → Ⅎ𝑦𝜑)
113, 7, 8, 10wl-eudf 34873 1 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1536  wex 1781  wnf 1785  ∃!weu 2654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-mo 2624  df-eu 2655
This theorem is referenced by: (None)
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