Metamath Proof Explorer


Theorem dvelimalcased

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

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

Proof

Step Hyp Ref Expression
1 dvelimalcased.1 𝑥 𝜑
2 dvelimalcased.2 ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑧 𝜑 )
3 dvelimalcased.3 ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝜓 )
4 dvelimalcased.4 ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑧 𝜃 )
5 dvelimalcased.5 ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ( 𝑧 = 𝑥 → ( 𝜓𝜃 ) ) )
6 dvelimalcased.6 ( ( 𝜑 ∧ ∀ 𝑥 𝑥 = 𝑦 ) → ( 𝜒𝜃 ) )
7 dvelimalcased.7 ( 𝜑 → ∀ 𝑧 𝜓 )
8 dvelimalcased.8 ( 𝜑 → ∀ 𝑥 𝜒 )
9 nfa1 𝑥𝑥 𝑥 = 𝑦
10 1 9 nfan 𝑥 ( 𝜑 ∧ ∀ 𝑥 𝑥 = 𝑦 )
11 10 6 alimd ( ( 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 cbv1v ( ( 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 )