Theorem dvelimexcasei 35754

Description: Eliminate a disjoint variable condition from an existentially quantified statement using cases. Inference form of dvelimexcased 35753. See axnulg 35869 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 1836	. . 3 ⊢ Ⅎ𝑥⊤
2		nfvd 1947	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧⊤)
3		dvelimexcasei.1	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
4	3	adantl 487	. . 3 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜑)
5		dvelimexcasei.2	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜒)
6	5	adantl 487	. . 3 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜒)
7		dvelimexcasei.3	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝜑 → 𝜒)))
8	7	adantl 487	. . 3 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑧 = 𝑥 → (𝜑 → 𝜒)))
9		dvelimexcasei.4	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜓 → 𝜒))
10	9	adantl 487	. . 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 35753	. 2 ⊢ (⊤ → ∃𝑥𝜒)
16	15	mptru 1576	1 ⊢ ∃𝑥𝜒

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1567  ⊤wtru 1570  ∃wex 1811  Ⅎwnf 1815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-11 2193  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816

This theorem is used by:  axsepg3  35865  axsepg3ALT  35866  axsepg5  35868  axnulg  35869