Theorem regsfromsetind 36669

Description: Derivation of ax-regs 35282 from mh-setind 36666. (Contributed by Matthew House, 4-Mar-2026.)

Hypothesis

Ref	Expression
regsfromsetind.1	⊢ (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) → ¬ 𝜑)

Assertion

Ref	Expression
regsfromsetind	⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))

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

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem regsfromsetind

Step	Hyp	Ref	Expression
1		nfia1 2158	. . . . 5 ⊢ Ⅎ𝑥(∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))
2	1	nfal 2328	. . . 4 ⊢ Ⅎ𝑥∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))
3	2	nfn 1858	. . 3 ⊢ Ⅎ𝑥 ¬ ∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))
4		regsfromsetind.1	. . . 4 ⊢ (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) → ¬ 𝜑)
5	4	con2i 139	. . 3 ⊢ (𝜑 → ¬ ∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
6	3, 5	exlimi 2224	. 2 ⊢ (∃𝑥𝜑 → ¬ ∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
7		exnalimn 1845	. . 3 ⊢ (∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))) ↔ ¬ ∀𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) → ¬ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
8		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑧(𝑥 ∈ 𝑦 → ¬ 𝜑)
9		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ∈ 𝑦
10		nfna1 2157	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)
11	9, 10	nfim 1897	. . . . . . 7 ⊢ Ⅎ𝑥(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))
12		elequ1 2120	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦))
13		sbequ12 2258	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
14		sb6 2090	. . . . . . . . . 10 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑))
15	13, 14	bitrdi 287	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
16	15	notbid 318	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (¬ 𝜑 ↔ ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
17	12, 16	imbi12d 344	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝑦 → ¬ 𝜑) ↔ (𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
18	8, 11, 17	cbvalv1 2345	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝜑) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
19		alinexa 1844	. . . . . . 7 ⊢ (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
20		sbalex 2249	. . . . . . 7 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
21	19, 20	xchbinx 334	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑))
22	18, 21	imbi12i 350	. . . . 5 ⊢ ((∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ (∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)) → ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
23		con2b 359	. . . . 5 ⊢ ((∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)) → ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ¬ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
24	22, 23	bitri 275	. . . 4 ⊢ ((∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ¬ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
25	24	albii 1820	. . 3 ⊢ (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ ∀𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) → ¬ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
26	7, 25	xchbinxr 335	. 2 ⊢ (∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))) ↔ ¬ ∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
27	6, 26	sylibr 234	1 ⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1539  ∃wex 1780  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-10 2146  ax-11 2162  ax-12 2184

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-nf 1785  df-sb 2068

This theorem is referenced by: (None)