MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvelimh Structured version   Visualization version   GIF version

Theorem dvelimh 2324
Description: Version of dvelim 2325 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 2011 . . 3 𝑥𝜑
3 dvelimh.2 . . . 4 (𝜓 → ∀𝑧𝜓)
43nf5i 2011 . . 3 𝑧𝜓
5 dvelimh.3 . . 3 (𝑧 = 𝑦 → (𝜑𝜓))
62, 4, 5dvelimf 2322 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
76nf5rd 2054 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wal 1473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701
This theorem is referenced by:  dvelim  2325  dveeq1-o16  33239  dveel2ALT  33242  ax6e2nd  37795  ax6e2ndVD  38166  ax6e2ndALT  38188
  Copyright terms: Public domain W3C validator