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Theorem dvelimdf 2334
Description: Deduction form of dvelimf 2333. (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.1 . . . 4 𝑥𝜑
2 dvelimdf.3 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
31, 2nfim1 2065 . . 3 𝑥(𝜑𝜓)
4 dvelimdf.2 . . . 4 𝑧𝜑
5 dvelimdf.4 . . . 4 (𝜑 → Ⅎ𝑧𝜒)
64, 5nfim1 2065 . . 3 𝑧(𝜑𝜒)
7 dvelimdf.5 . . . . 5 (𝜑 → (𝑧 = 𝑦 → (𝜓𝜒)))
87com12 32 . . . 4 (𝑧 = 𝑦 → (𝜑 → (𝜓𝜒)))
98pm5.74d 262 . . 3 (𝑧 = 𝑦 → ((𝜑𝜓) ↔ (𝜑𝜒)))
103, 6, 9dvelimf 2333 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝜑𝜒))
11 pm5.5 351 . . 3 (𝜑 → ((𝜑𝜒) ↔ 𝜒))
121, 11nfbidf 2090 . 2 (𝜑 → (Ⅎ𝑥(𝜑𝜒) ↔ Ⅎ𝑥𝜒))
1310, 12syl5ib 234 1 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1478  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707
This theorem is referenced by:  nfsb4t  2388  dvelimdc  2782
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