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Theorem dvelimf 2332
Description: Version of dvelimv 2336 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 2289 . . 3 (∀𝑧(𝑧 = 𝑦𝜑) ↔ 𝜓)
43bicomi 214 . 2 (𝜓 ↔ ∀𝑧(𝑧 = 𝑦𝜑))
5 nfnae 2316 . . 3 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
6 nfeqf 2299 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑧 = 𝑦)
76ancoms 469 . . . 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 1821 . . 3 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑧 = 𝑦𝜑))
115, 10nfald2 2329 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑧(𝑧 = 𝑦𝜑))
124, 11nfxfrd 1778 1 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1479  wnf 1706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708
This theorem is referenced by:  dvelimdf  2333  dvelimh  2334  dvelimnf  2337
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