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Theorem wl-mo2tf 34693
Description: Closed form of mof 2645 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 2453 . . 3 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2 nfa1 2148 . . 3 𝑥𝑥𝑦𝜑
31, 2nfan 1893 . 2 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑)
4 nfnae 2453 . . 3 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
5 nfnf1 2151 . . . 4 𝑦𝑦𝜑
65nfal 2336 . . 3 𝑦𝑥𝑦𝜑
74, 6nfan 1893 . 2 𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑)
8 simpl 483 . 2 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → ¬ ∀𝑥 𝑥 = 𝑦)
9 sp 2174 . . 3 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝜑)
109adantl 482 . 2 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → Ⅎ𝑦𝜑)
113, 7, 8, 10wl-mo2df 34692 1 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wal 1528  wex 1773  wnf 1777  ∃*wmo 2618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-mo 2620
This theorem is referenced by: (None)
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