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

Theorem dvelimf 2474
Description: Version of dvelimv 2478 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 2436 . . 3 (∀𝑧(𝑧 = 𝑦𝜑) ↔ 𝜓)
43bicomi 214 . 2 (𝜓 ↔ ∀𝑧(𝑧 = 𝑦𝜑))
5 nfnae 2460 . . 3 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
6 nfeqf 2446 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑧 = 𝑦)
76ancoms 468 . . . 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 1972 . . 3 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑧 = 𝑦𝜑))
115, 10nfald2 2471 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑧(𝑧 = 𝑦𝜑))
124, 11nfxfrd 1929 1 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1630  wnf 1857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859
This theorem is referenced by:  dvelimdf  2475  dvelimh  2476  dvelimnf  2479
  Copyright terms: Public domain W3C validator