Theorem dvelimexcased 35045

Description: Eliminate a disjoint variable condition from an existentially quantified statement using cases. (Contributed by BTernaryTau, 31-Jul-2025.)

Hypotheses

Ref	Expression
dvelimexcased.1	⊢ Ⅎ𝑥𝜑
dvelimexcased.2	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
dvelimexcased.3	⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
dvelimexcased.4	⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜃)
dvelimexcased.5	⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑧 = 𝑥 → (𝜓 → 𝜃)))
dvelimexcased.6	⊢ ((𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (𝜒 → 𝜃))
dvelimexcased.7	⊢ (𝜑 → ∃𝑧𝜓)
dvelimexcased.8	⊢ (𝜑 → ∃𝑥𝜒)

Assertion

Ref	Expression
dvelimexcased	⊢ (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 dvelimexcased

Step	Hyp	Ref	Expression
1		dvelimexcased.8	. . 3 ⊢ (𝜑 → ∃𝑥𝜒)
2		dvelimexcased.1	. . . . . 6 ⊢ Ⅎ𝑥𝜑
3		nfa1 2147	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 𝑥 = 𝑦
4	2, 3	nfan 1898	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑥 𝑥 = 𝑦)
5		dvelimexcased.6	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (𝜒 → 𝜃))
6	4, 5	eximd 2212	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (∃𝑥𝜒 → ∃𝑥𝜃))
7	6	ex 412	. . 3 ⊢ (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜒 → ∃𝑥𝜃)))
8	1, 7	mpid 44	. 2 ⊢ (𝜑 → (∀𝑥 𝑥 = 𝑦 → ∃𝑥𝜃))
9		dvelimexcased.7	. . 3 ⊢ (𝜑 → ∃𝑧𝜓)
10		nfv 1913	. . . . . 6 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
11		dvelimexcased.2	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
12	10, 11	nfan1c 35041	. . . . 5 ⊢ Ⅎ𝑧(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
13		nfna1 2148	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
14	2, 13	nfan 1898	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
15		dvelimexcased.3	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
16		dvelimexcased.4	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜃)
17		dvelimexcased.5	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑧 = 𝑥 → (𝜓 → 𝜃)))
18	12, 14, 15, 16, 17	cbvex1v 35042	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∃𝑧𝜓 → ∃𝑥𝜃))
19	18	ex 412	. . 3 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜓 → ∃𝑥𝜃)))
20	9, 19	mpid 44	. 2 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝜃))
21	8, 20	pm2.61d 179	1 ⊢ (𝜑 → ∃𝑥𝜃)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1535  ∃wex 1777  Ⅎwnf 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:  dvelimexcasei  35046