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Theorem wl-mo2df 34987
 Description: Version of mof 2622 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate disjoint variable conditions. (Contributed by Wolf Lammen, 11-Aug-2019.)
Hypotheses
Ref Expression
wl-mo2df.1 𝑥𝜑
wl-mo2df.2 𝑦𝜑
wl-mo2df.3 (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦)
wl-mo2df.4 (𝜑 → Ⅎ𝑦𝜓)
Assertion
Ref Expression
wl-mo2df (𝜑 → (∃*𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦)))

Proof of Theorem wl-mo2df
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 df-mo 2598 . 2 (∃*𝑥𝜓 ↔ ∃𝑢𝑥(𝜓𝑥 = 𝑢))
2 wl-mo2df.2 . . 3 𝑦𝜑
3 wl-mo2df.1 . . . 4 𝑥𝜑
4 wl-mo2df.4 . . . . 5 (𝜑 → Ⅎ𝑦𝜓)
5 wl-mo2df.3 . . . . . 6 (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦)
6 nfeqf1 2386 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥 = 𝑢)
76naecoms 2440 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 = 𝑢)
85, 7syl 17 . . . . 5 (𝜑 → Ⅎ𝑦 𝑥 = 𝑢)
94, 8nfimd 1895 . . . 4 (𝜑 → Ⅎ𝑦(𝜓𝑥 = 𝑢))
103, 9nfald 2336 . . 3 (𝜑 → Ⅎ𝑦𝑥(𝜓𝑥 = 𝑢))
11 nfnae 2445 . . . . . . 7 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
12 nfeqf2 2384 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑢 = 𝑦)
1311, 12nfan1 2198 . . . . . 6 𝑥(¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑦)
14 equequ2 2033 . . . . . . . 8 (𝑢 = 𝑦 → (𝑥 = 𝑢𝑥 = 𝑦))
1514imbi2d 344 . . . . . . 7 (𝑢 = 𝑦 → ((𝜓𝑥 = 𝑢) ↔ (𝜓𝑥 = 𝑦)))
1615adantl 485 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑦) → ((𝜓𝑥 = 𝑢) ↔ (𝜓𝑥 = 𝑦)))
1713, 16albid 2222 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑦) → (∀𝑥(𝜓𝑥 = 𝑢) ↔ ∀𝑥(𝜓𝑥 = 𝑦)))
185, 17sylan 583 . . . 4 ((𝜑𝑢 = 𝑦) → (∀𝑥(𝜓𝑥 = 𝑢) ↔ ∀𝑥(𝜓𝑥 = 𝑦)))
1918ex 416 . . 3 (𝜑 → (𝑢 = 𝑦 → (∀𝑥(𝜓𝑥 = 𝑢) ↔ ∀𝑥(𝜓𝑥 = 𝑦))))
202, 10, 19cbvexd 2418 . 2 (𝜑 → (∃𝑢𝑥(𝜓𝑥 = 𝑢) ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦)))
211, 20syl5bb 286 1 (𝜑 → (∃*𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  Ⅎwnf 1785  ∃*wmo 2596 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 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379 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 2598 This theorem is referenced by:  wl-mo2tf  34988
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