Theorem mh-regprimbi 36743

Description: Shortest possible version of ax-reg 9500 in primitive symbols. The equivalence is nontrivial, but it still follows solely from the axioms of predicate calculus. (Contributed by Matthew House, 13-Apr-2026.)

Assertion

Ref	Expression
mh-regprimbi	⊢ ((∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) ↔ ¬ ∀𝑦 ¬ ∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥))

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem mh-regprimbi

Step	Hyp	Ref	Expression
1		elequ1 2121	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
2	1	cbvexvw 2039	. . . 4 ⊢ (∃𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑧 𝑧 ∈ 𝑥)
3		df-ex 1782	. . . 4 ⊢ (∃𝑧 𝑧 ∈ 𝑥 ↔ ¬ ∀𝑧 ¬ 𝑧 ∈ 𝑥)
4	2, 3	bitri 275	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝑥 ↔ ¬ ∀𝑧 ¬ 𝑧 ∈ 𝑥)
5	4	imbi1i 349	. 2 ⊢ ((∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) ↔ (¬ ∀𝑧 ¬ 𝑧 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))))
6		jarl 125	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥) → (¬ 𝑦 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑥))
7	6	com12 32	. . . . . . . . . 10 ⊢ (¬ 𝑦 ∈ 𝑥 → (((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥) → ¬ 𝑧 ∈ 𝑥))
8	7	alimdv 1918	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ 𝑥 → (∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥) → ∀𝑧 ¬ 𝑧 ∈ 𝑥))
9	8	con3rr3 155	. . . . . . . 8 ⊢ (¬ ∀𝑧 ¬ 𝑧 ∈ 𝑥 → (¬ 𝑦 ∈ 𝑥 → ¬ ∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥)))
10	9	con4d 115	. . . . . . 7 ⊢ (¬ ∀𝑧 ¬ 𝑧 ∈ 𝑥 → (∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥) → 𝑦 ∈ 𝑥))
11	10	pm4.71rd 562	. . . . . 6 ⊢ (¬ ∀𝑧 ¬ 𝑧 ∈ 𝑥 → (∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥) ↔ (𝑦 ∈ 𝑥 ∧ ∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥))))
12		pm5.5 361	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑥 → ((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) ↔ 𝑧 ∈ 𝑦))
13	12	imbi1d 341	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑥 → (((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥) ↔ (𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
14	13	albidv 1922	. . . . . . 7 ⊢ (𝑦 ∈ 𝑥 → (∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
15	14	pm5.32i 574	. . . . . 6 ⊢ ((𝑦 ∈ 𝑥 ∧ ∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥)) ↔ (𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
16	11, 15	bitr2di 288	. . . . 5 ⊢ (¬ ∀𝑧 ¬ 𝑧 ∈ 𝑥 → ((𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)) ↔ ∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥)))
17	16	exbidv 1923	. . . 4 ⊢ (¬ ∀𝑧 ¬ 𝑧 ∈ 𝑥 → (∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)) ↔ ∃𝑦∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥)))
18	17	pm5.74i 271	. . 3 ⊢ ((¬ ∀𝑧 ¬ 𝑧 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) ↔ (¬ ∀𝑧 ¬ 𝑧 ∈ 𝑥 → ∃𝑦∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥)))
19		ala1 1815	. . . . . 6 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑥 → ∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥))
20	19	alrimiv 1929	. . . . 5 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑥 → ∀𝑦∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥))
21	20	19.2d 1979	. . . 4 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑥 → ∃𝑦∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥))
22	21	biantrur 530	. . 3 ⊢ ((¬ ∀𝑧 ¬ 𝑧 ∈ 𝑥 → ∃𝑦∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥)) ↔ ((∀𝑧 ¬ 𝑧 ∈ 𝑥 → ∃𝑦∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥)) ∧ (¬ ∀𝑧 ¬ 𝑧 ∈ 𝑥 → ∃𝑦∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥))))
23		pm4.83 1027	. . 3 ⊢ (((∀𝑧 ¬ 𝑧 ∈ 𝑥 → ∃𝑦∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥)) ∧ (¬ ∀𝑧 ¬ 𝑧 ∈ 𝑥 → ∃𝑦∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥))) ↔ ∃𝑦∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥))
24	18, 22, 23	3bitri 297	. 2 ⊢ ((¬ ∀𝑧 ¬ 𝑧 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) ↔ ∃𝑦∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥))
25		df-ex 1782	. 2 ⊢ (∃𝑦∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥) ↔ ¬ ∀𝑦 ¬ ∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥))
26	5, 24, 25	3bitri 297	1 ⊢ ((∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) ↔ ¬ ∀𝑦 ¬ ∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀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)