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Metamath Proof Explorer

Theorem dvelimf

Description: Version of dvelimv without any variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 1-Oct-2002) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Wolf Lammen, 11-May-2018) (New usage is discouraged.)

Ref Expression
Hypotheses dvelimf.1 ⊢ Ⅎ 𝑥 𝜑
dvelimf.2 ⊢ Ⅎ 𝑧 𝜓
dvelimf.3 ⊢ ( 𝑧 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion dvelimf ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝜓 )

Proof

Step Hyp Ref Expression
1 dvelimf.1 ⊢ Ⅎ 𝑥 𝜑
2 dvelimf.2 ⊢ Ⅎ 𝑧 𝜓
3 dvelimf.3 ⊢ ( 𝑧 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
4 2 3 equsal ⊢ ( ∀ 𝑧 ( 𝑧 = 𝑦 → 𝜑 ) ↔ 𝜓 )
5 4 bicomi ⊢ ( 𝜓 ↔ ∀ 𝑧 ( 𝑧 = 𝑦 → 𝜑 ) )
6 nfnae ⊢ Ⅎ 𝑧 ¬ ∀ 𝑥 𝑥 = 𝑦
7 nfeqf ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝑧 = 𝑦 )
8 7 ancoms ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 𝑧 = 𝑦 )
9 1 a1i ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 𝜑 )
10 8 9 nfimd ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → Ⅎ 𝑥 ( 𝑧 = 𝑦 → 𝜑 ) )
11 6 10 nfald2 ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 ∀ 𝑧 ( 𝑧 = 𝑦 → 𝜑 ) )
12 5 11 nfxfrd ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝜓 )