Theorem dveeq2or 1739

Description: Quantifier introduction when one pair of variables is distinct. Like dveeq2 1738 but connecting ∀𝑥𝑥 = 𝑦 by a disjunction rather than negation and implication makes the theorem stronger in intuitionistic logic. (Contributed by Jim Kingdon, 1-Feb-2018.)

Assertion

Ref	Expression
dveeq2or	⊢ (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥 𝑧 = 𝑦)

Distinct variable group:   𝑥,𝑧

Proof of Theorem dveeq2or

Step	Hyp	Ref	Expression
1		ax-i12 1439	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)))
2		orass 717	. . . . . 6 ⊢ (((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦) ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ↔ (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))))
3	1, 2	mpbir 144	. . . . 5 ⊢ ((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦) ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
4		pm1.4 679	. . . . . 6 ⊢ ((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦) → (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))
5	4	orim1i 710	. . . . 5 ⊢ (((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦) ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) → ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧) ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)))
6	3, 5	ax-mp 7	. . . 4 ⊢ ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧) ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
7		orass 717	. . . 4 ⊢ (((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧) ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ↔ (∀𝑥 𝑥 = 𝑦 ∨ (∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))))
8	6, 7	mpbi 143	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 ∨ (∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)))
9		ax16 1736	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑧 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
10	9	a5i 1476	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
11		id 19	. . . . 5 ⊢ (∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) → ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
12	10, 11	jaoi 669	. . . 4 ⊢ ((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) → ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
13	12	orim2i 711	. . 3 ⊢ ((∀𝑥 𝑥 = 𝑦 ∨ (∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))) → (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)))
14	8, 13	ax-mp 7	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
15		df-nf 1391	. . . 4 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 ↔ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
16	15	biimpri 131	. . 3 ⊢ (∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) → Ⅎ𝑥 𝑧 = 𝑦)
17	16	orim2i 711	. 2 ⊢ ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) → (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥 𝑧 = 𝑦))
18	14, 17	ax-mp 7	1 ⊢ (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥 𝑧 = 𝑦)

Syntax hints:   → wi 4   ∨ wo 662  ∀wal 1283  Ⅎwnf 1390

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688

This theorem is referenced by:  equs5or  1753  sbal1yz  1920  copsexg  4035