Theorem dvelimexcasei 35056

Description: Eliminate a disjoint variable condition from an existentially quantified statement using cases. Inference form of dvelimexcased 35055. See axnulg 35070 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 → (𝜑 → 𝜒)))
dvelimexcasei.4	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜓 → 𝜒))
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

Step	Hyp	Ref	Expression
1		nftru 1802	. . 3 ⊢ Ⅎ𝑥⊤
2		nfvd 1914	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧⊤)
3		dvelimexcasei.1	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
4	3	adantl 481	. . 3 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜑)
5		dvelimexcasei.2	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜒)
6	5	adantl 481	. . 3 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜒)
7		dvelimexcasei.3	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝜑 → 𝜒)))
8	7	adantl 481	. . 3 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑧 = 𝑥 → (𝜑 → 𝜒)))
9		dvelimexcasei.4	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜓 → 𝜒))
10	9	adantl 481	. . 3 ⊢ ((⊤ ∧ ∀𝑥 𝑥 = 𝑦) → (𝜓 → 𝜒))
11		dvelimexcasei.5	. . . 4 ⊢ ∃𝑧𝜑
12	11	a1i 11	. . 3 ⊢ (⊤ → ∃𝑧𝜑)
13		dvelimexcasei.6	. . . 4 ⊢ ∃𝑥𝜓
14	13	a1i 11	. . 3 ⊢ (⊤ → ∃𝑥𝜓)
15	1, 2, 4, 6, 8, 10, 12, 14	dvelimexcased 35055	. 2 ⊢ (⊤ → ∃𝑥𝜒)
16	15	mptru 1544	1 ⊢ ∃𝑥𝜒

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535  ⊤wtru 1538  ∃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 2141  ax-11 2158  ax-12 2178

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  35060  axsepg2ALT  35061  axnulg  35070