Theorem mh-prprimbi 36741

Description: Shortest possible version of ax-pr 5370 in primitive symbols. (Contributed by Matthew House, 13-Apr-2026.)

Assertion

Ref	Expression
mh-prprimbi	⊢ (∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) ↔ ¬ ∀𝑧(𝑥 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑧))

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

Proof of Theorem mh-prprimbi

Step	Hyp	Ref	Expression
1		jaob 964	. . . . 5 ⊢ (((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) ↔ ((𝑤 = 𝑥 → 𝑤 ∈ 𝑧) ∧ (𝑤 = 𝑦 → 𝑤 ∈ 𝑧)))
2	1	albii 1821	. . . 4 ⊢ (∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) ↔ ∀𝑤((𝑤 = 𝑥 → 𝑤 ∈ 𝑧) ∧ (𝑤 = 𝑦 → 𝑤 ∈ 𝑧)))
3		19.26 1872	. . . 4 ⊢ (∀𝑤((𝑤 = 𝑥 → 𝑤 ∈ 𝑧) ∧ (𝑤 = 𝑦 → 𝑤 ∈ 𝑧)) ↔ (∀𝑤(𝑤 = 𝑥 → 𝑤 ∈ 𝑧) ∧ ∀𝑤(𝑤 = 𝑦 → 𝑤 ∈ 𝑧)))
4		elequ1 2121	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧))
5	4	equsalvw 2006	. . . . 5 ⊢ (∀𝑤(𝑤 = 𝑥 → 𝑤 ∈ 𝑧) ↔ 𝑥 ∈ 𝑧)
6		elequ1 2121	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
7	6	equsalvw 2006	. . . . 5 ⊢ (∀𝑤(𝑤 = 𝑦 → 𝑤 ∈ 𝑧) ↔ 𝑦 ∈ 𝑧)
8	5, 7	anbi12i 629	. . . 4 ⊢ ((∀𝑤(𝑤 = 𝑥 → 𝑤 ∈ 𝑧) ∧ ∀𝑤(𝑤 = 𝑦 → 𝑤 ∈ 𝑧)) ↔ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧))
9	2, 3, 8	3bitri 297	. . 3 ⊢ (∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) ↔ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧))
10	9	exbii 1850	. 2 ⊢ (∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) ↔ ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧))
11		exnalimn 1846	. 2 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧) ↔ ¬ ∀𝑧(𝑥 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑧))
12	10, 11	bitri 275	1 ⊢ (∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) ↔ ¬ ∀𝑧(𝑥 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848  ∀wal 1540  ∃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 1912  ax-6 1969  ax-7 2010  ax-8 2116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782

This theorem is referenced by: (None)