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Theorem dvelimf 2428
Description: Version of dvelimv 2432 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)
Hypotheses
Ref Expression
dvelimf.1 𝑥𝜑
dvelimf.2 𝑧𝜓
dvelimf.3 (𝑧 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
dvelimf (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)

Proof of Theorem dvelimf
StepHypRef Expression
1 dvelimf.2 . . . 4 𝑧𝜓
2 dvelimf.3 . . . 4 (𝑧 = 𝑦 → (𝜑𝜓))
31, 2equsal 2390 . . 3 (∀𝑧(𝑧 = 𝑦𝜑) ↔ 𝜓)
43bicomi 215 . 2 (𝜓 ↔ ∀𝑧(𝑧 = 𝑦𝜑))
5 nfnae 2414 . . 3 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
6 nfeqf 2401 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑧 = 𝑦)
76ancoms 450 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑧 = 𝑦)
8 dvelimf.1 . . . . 5 𝑥𝜑
98a1i 11 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝜑)
107, 9nfimd 1992 . . 3 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑧 = 𝑦𝜑))
115, 10nfald2 2425 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑧(𝑧 = 𝑦𝜑))
124, 11nfxfrd 1949 1 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wal 1650  wnf 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879
This theorem is referenced by:  dvelimdf  2429  dvelimh  2430  dvelimnf  2433
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