Theorem dvelimf 2334

Description: Version of dvelimv 2338 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)

Hypotheses

Ref	Expression
dvelimf.1	⊢ Ⅎ𝑥𝜑
dvelimf.2	⊢ Ⅎ𝑧𝜓
dvelimf.3	⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
dvelimf	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)

Proof of Theorem dvelimf

Step	Hyp	Ref	Expression
1		dvelimf.2	. . . 4 ⊢ Ⅎ𝑧𝜓
2		dvelimf.3	. . . 4 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	equsal 2291	. . 3 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜑) ↔ 𝜓)
4	3	bicomi 214	. 2 ⊢ (𝜓 ↔ ∀𝑧(𝑧 = 𝑦 → 𝜑))
5		nfnae 2318	. . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
6		nfeqf 2301	. . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑧 = 𝑦)
7	6	ancoms 469	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑧 = 𝑦)
8		dvelimf.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
9	8	a1i 11	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝜑)
10	7, 9	nfimd 1823	. . 3 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑧 = 𝑦 → 𝜑))
11	5, 10	nfald2 2331	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑧(𝑧 = 𝑦 → 𝜑))
12	4, 11	nfxfrd 1780	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  dvelimdf  2335  dvelimh  2336  dvelimnf  2339