Metamath Proof Explorer


Theorem dvelimexcased

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

Ref Expression
Hypotheses dvelimexcased.1 𝑥 𝜑
dvelimexcased.2 ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑧 𝜑 )
dvelimexcased.3 ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝜓 )
dvelimexcased.4 ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑧 𝜃 )
dvelimexcased.5 ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ( 𝑧 = 𝑥 → ( 𝜓𝜃 ) ) )
dvelimexcased.6 ( ( 𝜑 ∧ ∀ 𝑥 𝑥 = 𝑦 ) → ( 𝜒𝜃 ) )
dvelimexcased.7 ( 𝜑 → ∃ 𝑧 𝜓 )
dvelimexcased.8 ( 𝜑 → ∃ 𝑥 𝜒 )
Assertion dvelimexcased ( 𝜑 → ∃ 𝑥 𝜃 )

Proof

Step Hyp Ref Expression
1 dvelimexcased.1 𝑥 𝜑
2 dvelimexcased.2 ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑧 𝜑 )
3 dvelimexcased.3 ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝜓 )
4 dvelimexcased.4 ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑧 𝜃 )
5 dvelimexcased.5 ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ( 𝑧 = 𝑥 → ( 𝜓𝜃 ) ) )
6 dvelimexcased.6 ( ( 𝜑 ∧ ∀ 𝑥 𝑥 = 𝑦 ) → ( 𝜒𝜃 ) )
7 dvelimexcased.7 ( 𝜑 → ∃ 𝑧 𝜓 )
8 dvelimexcased.8 ( 𝜑 → ∃ 𝑥 𝜒 )
9 nfa1 𝑥𝑥 𝑥 = 𝑦
10 1 9 nfan 𝑥 ( 𝜑 ∧ ∀ 𝑥 𝑥 = 𝑦 )
11 10 6 eximd ( ( 𝜑 ∧ ∀ 𝑥 𝑥 = 𝑦 ) → ( ∃ 𝑥 𝜒 → ∃ 𝑥 𝜃 ) )
12 11 ex ( 𝜑 → ( ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑥 𝜒 → ∃ 𝑥 𝜃 ) ) )
13 8 12 mpid ( 𝜑 → ( ∀ 𝑥 𝑥 = 𝑦 → ∃ 𝑥 𝜃 ) )
14 nfv 𝑧 ¬ ∀ 𝑥 𝑥 = 𝑦
15 14 2 nfan1c 𝑧 ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 )
16 nfna1 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑦
17 1 16 nfan 𝑥 ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 )
18 15 17 3 4 5 cbvex1v ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ( ∃ 𝑧 𝜓 → ∃ 𝑥 𝜃 ) )
19 18 ex ( 𝜑 → ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑧 𝜓 → ∃ 𝑥 𝜃 ) ) )
20 7 19 mpid ( 𝜑 → ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ∃ 𝑥 𝜃 ) )
21 13 20 pm2.61d ( 𝜑 → ∃ 𝑥 𝜃 )