Theorem dveeq2or 1869

Description: Quantifier introduction when one pair of variables is distinct. Like dveeq2 1868 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		ax12or 1561	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)))
2		orass 779	. . . . . 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 146	. . . . 5 ⊢ ((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦) ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
4		pm1.4 739	. . . . . 6 ⊢ ((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦) → (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))
5	4	orim1i 772	. . . . 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 5	. . . 4 ⊢ ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧) ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
7		orass 779	. . . 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 145	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 ∨ (∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)))
9		ax16 1866	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑧 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
10	9	a5i 1596	. . . . 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 728	. . . 4 ⊢ ((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) → ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
13	12	orim2i 773	. . 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 5	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
15		df-nf 1514	. . . 4 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 ↔ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
16	15	biimpri 133	. . 3 ⊢ (∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) → Ⅎ𝑥 𝑧 = 𝑦)
17	16	orim2i 773	. 2 ⊢ ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) → (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥 𝑧 = 𝑦))
18	14, 17	ax-mp 5	1 ⊢ (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥 𝑧 = 𝑦)

Syntax hints:   → wi 4   ∨ wo 720  ∀wal 1400  Ⅎwnf 1513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816

This theorem is referenced by:  equs5or  1883  sbal1yz  2061  copsexg  4379