Theorem dvelimor 1991

Description: Disjunctive distinct variable constraint elimination. A user of this theorem starts with a formula 𝜑 (containing 𝑧) and a distinct variable constraint between 𝑥 and 𝑧. The theorem makes it possible to replace the distinct variable constraint with the disjunct ∀𝑥𝑥 = 𝑦 (𝜓 is just a version of 𝜑 with 𝑦 substituted for 𝑧). (Contributed by Jim Kingdon, 11-May-2018.)

Hypotheses

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

Assertion

Ref	Expression
dvelimor	⊢ (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥𝜓)

Distinct variable groups:   𝜓,𝑧   𝑥,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦)

Proof of Theorem dvelimor

Step	Hyp	Ref	Expression
1		ax-bndl 1486	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑧∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)))
2		orcom 717	. . . . . . 7 ⊢ ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑧∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ↔ (∀𝑧∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) ∨ ∀𝑥 𝑥 = 𝑦))
3	2	orbi2i 751	. . . . . 6 ⊢ ((∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑧∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))) ↔ (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑧∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) ∨ ∀𝑥 𝑥 = 𝑦)))
4	1, 3	mpbi 144	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑧∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) ∨ ∀𝑥 𝑥 = 𝑦))
5		orass 756	. . . . 5 ⊢ (((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑧∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ∨ ∀𝑥 𝑥 = 𝑦) ↔ (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑧∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) ∨ ∀𝑥 𝑥 = 𝑦)))
6	4, 5	mpbir 145	. . . 4 ⊢ ((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑧∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ∨ ∀𝑥 𝑥 = 𝑦)
7		nfae 1697	. . . . . . 7 ⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑧
8		a16nf 1838	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥(𝑧 = 𝑦 → 𝜑))
9	7, 8	alrimi 1502	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑧Ⅎ𝑥(𝑧 = 𝑦 → 𝜑))
10		df-nf 1437	. . . . . . . 8 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 ↔ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
11		id 19	. . . . . . . . 9 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
12		dvelimor.1	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝜑
13	12	a1i 9	. . . . . . . . 9 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 → Ⅎ𝑥𝜑)
14	11, 13	nfimd 1564	. . . . . . . 8 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 → Ⅎ𝑥(𝑧 = 𝑦 → 𝜑))
15	10, 14	sylbir 134	. . . . . . 7 ⊢ (∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) → Ⅎ𝑥(𝑧 = 𝑦 → 𝜑))
16	15	alimi 1431	. . . . . 6 ⊢ (∀𝑧∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) → ∀𝑧Ⅎ𝑥(𝑧 = 𝑦 → 𝜑))
17	9, 16	jaoi 705	. . . . 5 ⊢ ((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑧∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) → ∀𝑧Ⅎ𝑥(𝑧 = 𝑦 → 𝜑))
18	17	orim1i 749	. . . 4 ⊢ (((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑧∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ∨ ∀𝑥 𝑥 = 𝑦) → (∀𝑧Ⅎ𝑥(𝑧 = 𝑦 → 𝜑) ∨ ∀𝑥 𝑥 = 𝑦))
19	6, 18	ax-mp 5	. . 3 ⊢ (∀𝑧Ⅎ𝑥(𝑧 = 𝑦 → 𝜑) ∨ ∀𝑥 𝑥 = 𝑦)
20		orcom 717	. . 3 ⊢ ((∀𝑧Ⅎ𝑥(𝑧 = 𝑦 → 𝜑) ∨ ∀𝑥 𝑥 = 𝑦) ↔ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑧Ⅎ𝑥(𝑧 = 𝑦 → 𝜑)))
21	19, 20	mpbi 144	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑧Ⅎ𝑥(𝑧 = 𝑦 → 𝜑))
22		nfalt 1557	. . . 4 ⊢ (∀𝑧Ⅎ𝑥(𝑧 = 𝑦 → 𝜑) → Ⅎ𝑥∀𝑧(𝑧 = 𝑦 → 𝜑))
23		ax-17 1506	. . . . . 6 ⊢ (𝜓 → ∀𝑧𝜓)
24		dvelimor.2	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))
25	23, 24	equsalh 1704	. . . . 5 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜑) ↔ 𝜓)
26	25	nfbii 1449	. . . 4 ⊢ (Ⅎ𝑥∀𝑧(𝑧 = 𝑦 → 𝜑) ↔ Ⅎ𝑥𝜓)
27	22, 26	sylib 121	. . 3 ⊢ (∀𝑧Ⅎ𝑥(𝑧 = 𝑦 → 𝜑) → Ⅎ𝑥𝜓)
28	27	orim2i 750	. 2 ⊢ ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑧Ⅎ𝑥(𝑧 = 𝑦 → 𝜑)) → (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥𝜓))
29	21, 28	ax-mp 5	1 ⊢ (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥𝜓)

Syntax hints:   → wi 4   ↔ wb 104   ∨ wo 697  ∀wal 1329  Ⅎwnf 1436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736

This theorem is referenced by:  nfsb4or  1996  rgen2a  2484