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Theorem dvelimalcased 35043
Description: Eliminate a disjoint variable condition from a universally quantified statement using cases. (Contributed by BTernaryTau, 31-Jul-2025.)
Hypotheses
Ref Expression
dvelimalcased.1 𝑥𝜑
dvelimalcased.2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
dvelimalcased.3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
dvelimalcased.4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜃)
dvelimalcased.5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑧 = 𝑥 → (𝜓𝜃)))
dvelimalcased.6 ((𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (𝜒𝜃))
dvelimalcased.7 (𝜑 → ∀𝑧𝜓)
dvelimalcased.8 (𝜑 → ∀𝑥𝜒)
Assertion
Ref Expression
dvelimalcased (𝜑 → ∀𝑥𝜃)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑥,𝑦,𝑧)   𝜃(𝑥,𝑦,𝑧)

Proof of Theorem dvelimalcased
StepHypRef Expression
1 dvelimalcased.8 . . 3 (𝜑 → ∀𝑥𝜒)
2 dvelimalcased.1 . . . . . 6 𝑥𝜑
3 nfa1 2147 . . . . . 6 𝑥𝑥 𝑥 = 𝑦
42, 3nfan 1898 . . . . 5 𝑥(𝜑 ∧ ∀𝑥 𝑥 = 𝑦)
5 dvelimalcased.6 . . . . 5 ((𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (𝜒𝜃))
64, 5alimd 2208 . . . 4 ((𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (∀𝑥𝜒 → ∀𝑥𝜃))
76ex 412 . . 3 (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜒 → ∀𝑥𝜃)))
81, 7mpid 44 . 2 (𝜑 → (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝜃))
9 dvelimalcased.7 . . 3 (𝜑 → ∀𝑧𝜓)
10 nfv 1913 . . . . . 6 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
11 dvelimalcased.2 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
1210, 11nfan1c 35041 . . . . 5 𝑧(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
13 nfna1 2148 . . . . . 6 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
142, 13nfan 1898 . . . . 5 𝑥(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
15 dvelimalcased.3 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
16 dvelimalcased.4 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜃)
17 dvelimalcased.5 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑧 = 𝑥 → (𝜓𝜃)))
1812, 14, 15, 16, 17cbv1v 2335 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑧𝜓 → ∀𝑥𝜃))
1918ex 412 . . 3 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜓 → ∀𝑥𝜃)))
209, 19mpid 44 . 2 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥𝜃))
218, 20pm2.61d 179 1 (𝜑 → ∀𝑥𝜃)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1535  wnf 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2136  ax-11 2153  ax-12 2173
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782
This theorem is referenced by:  dvelimalcasei  35044
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