Theorem mh-prprimbi 37373

Description: Shortest possible version of ax-pr 5390 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 976	. . . . 5 ⊢ (((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) ↔ ((𝑤 = 𝑥 → 𝑤 ∈ 𝑧) ∧ (𝑤 = 𝑦 → 𝑤 ∈ 𝑧)))
2	1	albii 1851	. . . 4 ⊢ (∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) ↔ ∀𝑤((𝑤 = 𝑥 → 𝑤 ∈ 𝑧) ∧ (𝑤 = 𝑦 → 𝑤 ∈ 𝑧)))
3		19.26 1902	. . . 4 ⊢ (∀𝑤((𝑤 = 𝑥 → 𝑤 ∈ 𝑧) ∧ (𝑤 = 𝑦 → 𝑤 ∈ 𝑧)) ↔ (∀𝑤(𝑤 = 𝑥 → 𝑤 ∈ 𝑧) ∧ ∀𝑤(𝑤 = 𝑦 → 𝑤 ∈ 𝑧)))
4		elequ1 2151	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧))
5	4	equsalvw 2036	. . . . 5 ⊢ (∀𝑤(𝑤 = 𝑥 → 𝑤 ∈ 𝑧) ↔ 𝑥 ∈ 𝑧)
6		elequ1 2151	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
7	6	equsalvw 2036	. . . . 5 ⊢ (∀𝑤(𝑤 = 𝑦 → 𝑤 ∈ 𝑧) ↔ 𝑦 ∈ 𝑧)
8	5, 7	anbi12i 640	. . . 4 ⊢ ((∀𝑤(𝑤 = 𝑥 → 𝑤 ∈ 𝑧) ∧ ∀𝑤(𝑤 = 𝑦 → 𝑤 ∈ 𝑧)) ↔ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧))
9	2, 3, 8	3bitri 300	. . 3 ⊢ (∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) ↔ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧))
10	9	exbii 1880	. 2 ⊢ (∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) ↔ ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧))
11		exnalimn 1876	. 2 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧) ↔ ¬ ∀𝑧(𝑥 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑧))
12	10, 11	bitri 278	1 ⊢ (∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) ↔ ¬ ∀𝑧(𝑥 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑧))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812

This theorem is used by: (None)