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Theorem dvelimh 2429
 Description: Version of dvelim 2430 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Proof shortened by Wolf Lammen, 11-May-2018.)
Hypotheses
Ref Expression
dvelimh.1 (𝜑 → ∀𝑥𝜑)
dvelimh.2 (𝜓 → ∀𝑧𝜓)
dvelimh.3 (𝑧 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
dvelimh (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Proof of Theorem dvelimh
StepHypRef Expression
1 dvelimh.1 . . . 4 (𝜑 → ∀𝑥𝜑)
21nf5i 2117 . . 3 𝑥𝜑
3 dvelimh.2 . . . 4 (𝜓 → ∀𝑧𝜓)
43nf5i 2117 . . 3 𝑧𝜓
5 dvelimh.3 . . 3 (𝑧 = 𝑦 → (𝜑𝜓))
62, 4, 5dvelimf 2427 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
76nf5rd 2161 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207  ∀wal 1520 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766 This theorem is referenced by:  dvelim  2430  dveeq1-o16  35629  dveel2ALT  35632  ax6e2nd  40457  ax6e2ndVD  40807  ax6e2ndALT  40829
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