ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dvelimdc GIF version

Theorem dvelimdc 2213
Description: Deduction form of dvelimc 2214. (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 1437 . . 3 𝑤(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
2 dvelimdc.1 . . . . 5 𝑥𝜑
3 dvelimdc.2 . . . . 5 𝑧𝜑
4 dvelimdc.3 . . . . . 6 (𝜑𝑥𝐴)
54nfcrd 2207 . . . . 5 (𝜑 → Ⅎ𝑥 𝑤𝐴)
6 dvelimdc.4 . . . . . 6 (𝜑𝑧𝐵)
76nfcrd 2207 . . . . 5 (𝜑 → Ⅎ𝑧 𝑤𝐵)
8 dvelimdc.5 . . . . . 6 (𝜑 → (𝑧 = 𝑦𝐴 = 𝐵))
9 eleq2 2117 . . . . . 6 (𝐴 = 𝐵 → (𝑤𝐴𝑤𝐵))
108, 9syl6 33 . . . . 5 (𝜑 → (𝑧 = 𝑦 → (𝑤𝐴𝑤𝐵)))
112, 3, 5, 7, 10dvelimdf 1908 . . . 4 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑤𝐵))
1211imp 119 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑤𝐵)
131, 12nfcd 2189 . 2 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝐵)
1413ex 112 1 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wnf 1365  wcel 1409  wnfc 2181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-nfc 2183
This theorem is referenced by:  dvelimc  2214
  Copyright terms: Public domain W3C validator