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

Theorem dvelimdc 2924
Description: Deduction form of dvelimc 2925. (Contributed by Mario Carneiro, 8-Oct-2016.)
Hypotheses
Ref Expression
dvelimdc.1 𝑥𝜑
dvelimdc.2 𝑧𝜑
dvelimdc.3 (𝜑𝑥𝐴)
dvelimdc.4 (𝜑𝑧𝐵)
dvelimdc.5 (𝜑 → (𝑧 = 𝑦𝐴 = 𝐵))
Assertion
Ref Expression
dvelimdc (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵))

Proof of Theorem dvelimdc
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfv 1992 . . 3 𝑤(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
2 dvelimdc.1 . . . . 5 𝑥𝜑
3 dvelimdc.2 . . . . 5 𝑧𝜑
4 dvelimdc.3 . . . . . 6 (𝜑𝑥𝐴)
54nfcrd 2909 . . . . 5 (𝜑 → Ⅎ𝑥 𝑤𝐴)
6 dvelimdc.4 . . . . . 6 (𝜑𝑧𝐵)
76nfcrd 2909 . . . . 5 (𝜑 → Ⅎ𝑧 𝑤𝐵)
8 dvelimdc.5 . . . . . 6 (𝜑 → (𝑧 = 𝑦𝐴 = 𝐵))
9 eleq2 2828 . . . . . 6 (𝐴 = 𝐵 → (𝑤𝐴𝑤𝐵))
108, 9syl6 35 . . . . 5 (𝜑 → (𝑧 = 𝑦 → (𝑤𝐴𝑤𝐵)))
112, 3, 5, 7, 10dvelimdf 2475 . . . 4 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑤𝐵))
1211imp 444 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑤𝐵)
131, 12nfcd 2897 . 2 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝐵)
1413ex 449 1 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wnf 1857  wcel 2139  wnfc 2889
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-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-cleq 2753  df-clel 2756  df-nfc 2891
This theorem is referenced by:  dvelimc  2925
  Copyright terms: Public domain W3C validator