Theorem eujustALT 2572

Description: Alternate proof of eujust 2571 illustrating the use of dvelim 2456. (Contributed by NM, 11-Mar-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
eujustALT	⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))

Distinct variable groups:   𝑥,𝑦   𝑥,𝑧   𝜑,𝑦   𝜑,𝑧

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eujustALT

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equequ2 2025	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
2	1	bibi2d 342	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝜑 ↔ 𝑥 = 𝑦) ↔ (𝜑 ↔ 𝑥 = 𝑧)))
3	2	albidv 1920	. . . 4 ⊢ (𝑦 = 𝑧 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
4	3	sps 2185	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
5	4	drex1 2446	. 2 ⊢ (∀𝑦 𝑦 = 𝑧 → (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
6		hbnae 2437	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ∀𝑦 ¬ ∀𝑦 𝑦 = 𝑧)
7		hbnae 2437	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧)
8	6, 7	alrimih 1824	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ∀𝑦∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧)
9		ax-5 1910	. . . . . . . 8 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤) → ∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤))
10		equequ2 2025	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (𝑥 = 𝑤 ↔ 𝑥 = 𝑦))
11	10	bibi2d 342	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → ((𝜑 ↔ 𝑥 = 𝑤) ↔ (𝜑 ↔ 𝑥 = 𝑦)))
12	11	albidv 1920	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
13	12	notbid 318	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
14	9, 13	dvelim 2456	. . . . . . 7 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
15	14	naecoms 2434	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
16		ax-5 1910	. . . . . . 7 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤) → ∀𝑦 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤))
17		equequ2 2025	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (𝑥 = 𝑤 ↔ 𝑥 = 𝑧))
18	17	bibi2d 342	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → ((𝜑 ↔ 𝑥 = 𝑤) ↔ (𝜑 ↔ 𝑥 = 𝑧)))
19	18	albidv 1920	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
20	19	notbid 318	. . . . . . 7 ⊢ (𝑤 = 𝑧 → (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
21	16, 20	dvelim 2456	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∀𝑦 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
22	3	notbid 318	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
23	22	a1i 11	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (𝑦 = 𝑧 → (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧))))
24	15, 21, 23	cbv2h 2411	. . . . 5 ⊢ (∀𝑦∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑦 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
25	8, 24	syl 17	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑦 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
26	25	notbid 318	. . 3 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑦 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ¬ ∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
27		df-ex 1780	. . 3 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ¬ ∀𝑦 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
28		df-ex 1780	. . 3 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ¬ ∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
29	26, 27, 28	3bitr4g 314	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
30	5, 29	pm2.61i 182	1 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784

This theorem is referenced by: (None)