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Theorem dvelimexcasei 35046
Description: Eliminate a disjoint variable condition from an existentially quantified statement using cases. Inference form of dvelimexcased 35045. See axnulg 35060 for an example of its use. (Contributed by BTernaryTau, 31-Jul-2025.)
Hypotheses
Ref Expression
dvelimexcasei.1 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
dvelimexcasei.2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜒)
dvelimexcasei.3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝜑𝜒)))
dvelimexcasei.4 (∀𝑥 𝑥 = 𝑦 → (𝜓𝜒))
dvelimexcasei.5 𝑧𝜑
dvelimexcasei.6 𝑥𝜓
Assertion
Ref Expression
dvelimexcasei 𝑥𝜒
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑥,𝑦,𝑧)

Proof of Theorem dvelimexcasei
StepHypRef Expression
1 nftru 1802 . . 3 𝑥
2 nfvd 1914 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧⊤)
3 dvelimexcasei.1 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
43adantl 481 . . 3 ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜑)
5 dvelimexcasei.2 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜒)
65adantl 481 . . 3 ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜒)
7 dvelimexcasei.3 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝜑𝜒)))
87adantl 481 . . 3 ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑧 = 𝑥 → (𝜑𝜒)))
9 dvelimexcasei.4 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝜓𝜒))
109adantl 481 . . 3 ((⊤ ∧ ∀𝑥 𝑥 = 𝑦) → (𝜓𝜒))
11 dvelimexcasei.5 . . . 4 𝑧𝜑
1211a1i 11 . . 3 (⊤ → ∃𝑧𝜑)
13 dvelimexcasei.6 . . . 4 𝑥𝜓
1413a1i 11 . . 3 (⊤ → ∃𝑥𝜓)
151, 2, 4, 6, 8, 10, 12, 14dvelimexcased 35045 . 2 (⊤ → ∃𝑥𝜒)
1615mptru 1544 1 𝑥𝜒
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1535  wtru 1538  wex 1777  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:  axsepg2  35050  axsepg2ALT  35051  axnulg  35060
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