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

Theorem dvelimdf 1892
 Description: Deduction form of dvelimf 1891. This version may be useful if we want to avoid ax-17 1419 and use ax-16 1695 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)
Hypotheses
Ref Expression
dvelimdf.1 𝑥𝜑
dvelimdf.2 𝑧𝜑
dvelimdf.3 (𝜑 → Ⅎ𝑥𝜓)
dvelimdf.4 (𝜑 → Ⅎ𝑧𝜒)
dvelimdf.5 (𝜑 → (𝑧 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
dvelimdf (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))

Proof of Theorem dvelimdf
StepHypRef Expression
1 dvelimdf.2 . . . 4 𝑧𝜑
2 dvelimdf.3 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
31, 2alrimi 1415 . . 3 (𝜑 → ∀𝑧𝑥𝜓)
4 nfsb4t 1890 . . 3 (∀𝑧𝑥𝜓 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑧]𝜓))
53, 4syl 14 . 2 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑧]𝜓))
6 dvelimdf.1 . . 3 𝑥𝜑
7 dvelimdf.4 . . . 4 (𝜑 → Ⅎ𝑧𝜒)
8 dvelimdf.5 . . . 4 (𝜑 → (𝑧 = 𝑦 → (𝜓𝜒)))
91, 7, 8sbied 1671 . . 3 (𝜑 → ([𝑦 / 𝑧]𝜓𝜒))
106, 9nfbidf 1432 . 2 (𝜑 → (Ⅎ𝑥[𝑦 / 𝑧]𝜓 ↔ Ⅎ𝑥𝜒))
115, 10sylibd 138 1 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  ∀wal 1241  Ⅎwnf 1349  [wsb 1645 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646 This theorem is referenced by:  dvelimdc  2197
 Copyright terms: Public domain W3C validator