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

Theorem dvelimnf 2327
Description: Version of dvelim 2325 using "not free" notation. (Contributed by Mario Carneiro, 9-Oct-2016.)
Hypotheses
Ref Expression
dvelimnf.1 𝑥𝜑
dvelimnf.2 (𝑧 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
dvelimnf (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
Distinct variable group:   𝜓,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦)

Proof of Theorem dvelimnf
StepHypRef Expression
1 dvelimnf.1 . 2 𝑥𝜑
2 nfv 1830 . 2 𝑧𝜓
3 dvelimnf.2 . 2 (𝑧 = 𝑦 → (𝜑𝜓))
41, 2, 3dvelimf 2322 1 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wal 1473  wnf 1699
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:  nfrab  3100
  Copyright terms: Public domain W3C validator