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Theorem wl-mo2tf 32532
 Description: Closed form of mo2 2467 with a distinctor avoiding distinct variable conditions. (Contributed by Wolf Lammen, 20-Sep-2020.)
Assertion
Ref Expression
wl-mo2tf ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))

Proof of Theorem wl-mo2tf
StepHypRef Expression
1 nfnae 2306 . . 3 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2 nfa1 2015 . . 3 𝑥𝑥𝑦𝜑
31, 2nfan 1816 . 2 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑)
4 nfnae 2306 . . 3 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
5 nfnf1 2018 . . . 4 𝑦𝑦𝜑
65nfal 2139 . . 3 𝑦𝑥𝑦𝜑
74, 6nfan 1816 . 2 𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑)
8 simpl 472 . 2 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → ¬ ∀𝑥 𝑥 = 𝑦)
9 sp 2041 . . 3 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝜑)
109adantl 481 . 2 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → Ⅎ𝑦𝜑)
113, 7, 8, 10wl-mo2df 32531 1 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699  ∃*wmo 2459 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463 This theorem is referenced by: (None)
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