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Theorem dvelimdf 1934
Description: Deduction form of dvelimf 1933. This version may be useful if we want to avoid ax-17 1460 and use ax-16 1736 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 1456 . . 3 (𝜑 → ∀𝑧𝑥𝜓)
4 nfsb4t 1932 . . 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 1712 . . 3 (𝜑 → ([𝑦 / 𝑧]𝜓𝜒))
106, 9nfbidf 1473 . 2 (𝜑 → (Ⅎ𝑥[𝑦 / 𝑧]𝜓 ↔ Ⅎ𝑥𝜒))
115, 10sylibd 147 1 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  wal 1283  wnf 1390  [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687
This theorem is referenced by:  dvelimdc  2239
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