Theorem eujustALT 2632

Description: Alternate proof of eujust 2631 illustrating the use of dvelim 2462. (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 2033	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
2	1	bibi2d 346	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝜑 ↔ 𝑥 = 𝑦) ↔ (𝜑 ↔ 𝑥 = 𝑧)))
3	2	albidv 1921	. . . 4 ⊢ (𝑦 = 𝑧 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
4	3	sps 2182	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
5	4	drex1 2452	. 2 ⊢ (∀𝑦 𝑦 = 𝑧 → (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
6		hbnae 2443	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ∀𝑦 ¬ ∀𝑦 𝑦 = 𝑧)
7		hbnae 2443	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧)
8	6, 7	alrimih 1825	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ∀𝑦∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧)
9		ax-5 1911	. . . . . . . 8 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤) → ∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤))
10		equequ2 2033	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (𝑥 = 𝑤 ↔ 𝑥 = 𝑦))
11	10	bibi2d 346	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → ((𝜑 ↔ 𝑥 = 𝑤) ↔ (𝜑 ↔ 𝑥 = 𝑦)))
12	11	albidv 1921	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
13	12	notbid 321	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
14	9, 13	dvelim 2462	. . . . . . 7 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
15	14	naecoms 2440	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
16		ax-5 1911	. . . . . . 7 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤) → ∀𝑦 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤))
17		equequ2 2033	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (𝑥 = 𝑤 ↔ 𝑥 = 𝑧))
18	17	bibi2d 346	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → ((𝜑 ↔ 𝑥 = 𝑤) ↔ (𝜑 ↔ 𝑥 = 𝑧)))
19	18	albidv 1921	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
20	19	notbid 321	. . . . . . 7 ⊢ (𝑤 = 𝑧 → (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
21	16, 20	dvelim 2462	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∀𝑦 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
22	3	notbid 321	. . . . . . 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 2415	. . . . 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 321	. . 3 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑦 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ¬ ∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
27		df-ex 1782	. . 3 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ¬ ∀𝑦 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
28		df-ex 1782	. . 3 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ¬ ∀𝑧 ¬ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
29	26, 27, 28	3bitr4g 317	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
30	5, 29	pm2.61i 185	1 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1536  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786

This theorem is referenced by: (None)