Theorem regsfromregtco 36993

Description: Derivation of ax-regs 35493 from ax-reg 9553 + ax-tco 36927. (Contributed by Matthew House, 4-Mar-2026.)

Hypotheses

Ref	Expression
regsfromregtco.1	⊢ (∃𝑦 𝑦 ∈ 𝑤 → ∃𝑦(𝑦 ∈ 𝑤 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑤)))
regsfromregtco.2	⊢ ∃𝑢(𝑣 ∈ 𝑢 ∧ ∀𝑡(𝑡 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑡 → 𝑠 ∈ 𝑢)))

Assertion

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

Distinct variable groups:   𝑤,𝑢,𝑥,𝑦,𝑧   𝑥,𝑣   𝑡,𝑠,𝑢   𝜑,𝑢,𝑤,𝑦,𝑧   𝜑,𝑣,𝑢,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑡,𝑠)

Proof of Theorem regsfromregtco

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3457	. . . . . . . . 9 ⊢ 𝑣 ∈ V
2		elequ1 2148	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (𝑦 ∈ 𝑢 ↔ 𝑣 ∈ 𝑢))
3		sbequ 2115	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → ([𝑦 / 𝑥]𝜑 ↔ [𝑣 / 𝑥]𝜑))
4	2, 3	anbi12d 643	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → ((𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝑣 ∈ 𝑢 ∧ [𝑣 / 𝑥]𝜑)))
5	1, 4	spcev 3564	. . . . . . . 8 ⊢ ((𝑣 ∈ 𝑢 ∧ [𝑣 / 𝑥]𝜑) → ∃𝑦(𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑))
6	5	adantlr 727	. . . . . . 7 ⊢ (((𝑣 ∈ 𝑢 ∧ Tr 𝑢) ∧ [𝑣 / 𝑥]𝜑) → ∃𝑦(𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑))
7		vex 3457	. . . . . . . . 9 ⊢ 𝑢 ∈ V
8	7	rabex 5309	. . . . . . . 8 ⊢ {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} ∈ V
9		eleq2 2850	. . . . . . . . . . 11 ⊢ (𝑤 = {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑}))
10		sbequ 2115	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑦 → ([𝑟 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
11	10	elrab 3649	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} ↔ (𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑))
12	9, 11	bitrdi 290	. . . . . . . . . 10 ⊢ (𝑤 = {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} → (𝑦 ∈ 𝑤 ↔ (𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑)))
13	12	exbidv 1949	. . . . . . . . 9 ⊢ (𝑤 = {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} → (∃𝑦 𝑦 ∈ 𝑤 ↔ ∃𝑦(𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑)))
14		eleq2 2850	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑}))
15		sbequ 2115	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑧 → ([𝑟 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
16	15	elrab 3649	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} ↔ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))
17	14, 16	bitrdi 290	. . . . . . . . . . . . . 14 ⊢ (𝑤 = {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} → (𝑧 ∈ 𝑤 ↔ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑)))
18	17	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑤 = {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} → (¬ 𝑧 ∈ 𝑤 ↔ ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑)))
19	18	imbi2d 343	. . . . . . . . . . . 12 ⊢ (𝑤 = {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} → ((𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑤) ↔ (𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))))
20	19	albidv 1948	. . . . . . . . . . 11 ⊢ (𝑤 = {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} → (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑤) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))))
21	12, 20	anbi12d 643	. . . . . . . . . 10 ⊢ (𝑤 = {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} → ((𝑦 ∈ 𝑤 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑤)) ↔ ((𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑)))))
22	21	exbidv 1949	. . . . . . . . 9 ⊢ (𝑤 = {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} → (∃𝑦(𝑦 ∈ 𝑤 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑤)) ↔ ∃𝑦((𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑)))))
23	13, 22	imbi12d 347	. . . . . . . 8 ⊢ (𝑤 = {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} → ((∃𝑦 𝑦 ∈ 𝑤 → ∃𝑦(𝑦 ∈ 𝑤 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑤))) ↔ (∃𝑦(𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑) → ∃𝑦((𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))))))
24		regsfromregtco.1	. . . . . . . 8 ⊢ (∃𝑦 𝑦 ∈ 𝑤 → ∃𝑦(𝑦 ∈ 𝑤 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑤)))
25	8, 23, 24	vtocl 3524	. . . . . . 7 ⊢ (∃𝑦(𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑) → ∃𝑦((𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))))
26	6, 25	syl 18	. . . . . 6 ⊢ (((𝑣 ∈ 𝑢 ∧ Tr 𝑢) ∧ [𝑣 / 𝑥]𝜑) → ∃𝑦((𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))))
27		imnan 404	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑢 → ¬ [𝑧 / 𝑥]𝜑) ↔ ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))
28		trel 5225	. . . . . . . . . . . . . . . . . 18 ⊢ (Tr 𝑢 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑢) → 𝑧 ∈ 𝑢))
29	28	imp 411	. . . . . . . . . . . . . . . . 17 ⊢ ((Tr 𝑢 ∧ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑢)) → 𝑧 ∈ 𝑢)
30	29	anass1rs 667	. . . . . . . . . . . . . . . 16 ⊢ (((Tr 𝑢 ∧ 𝑦 ∈ 𝑢) ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ 𝑢)
31		imbibi 395	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∈ 𝑢 → ¬ [𝑧 / 𝑥]𝜑) ↔ ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑)) → (𝑧 ∈ 𝑢 → (¬ [𝑧 / 𝑥]𝜑 ↔ ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))))
32	27, 30, 31	mpsyl 69	. . . . . . . . . . . . . . 15 ⊢ (((Tr 𝑢 ∧ 𝑦 ∈ 𝑢) ∧ 𝑧 ∈ 𝑦) → (¬ [𝑧 / 𝑥]𝜑 ↔ ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑)))
33	32	pm5.74da 815	. . . . . . . . . . . . . 14 ⊢ ((Tr 𝑢 ∧ 𝑦 ∈ 𝑢) → ((𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑) ↔ (𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))))
34	33	albidv 1948	. . . . . . . . . . . . 13 ⊢ ((Tr 𝑢 ∧ 𝑦 ∈ 𝑢) → (∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))))
35	34	biimpar 482	. . . . . . . . . . . 12 ⊢ (((Tr 𝑢 ∧ 𝑦 ∈ 𝑢) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))) → ∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑))
36	35	anim2i 628	. . . . . . . . . . 11 ⊢ (([𝑦 / 𝑥]𝜑 ∧ ((Tr 𝑢 ∧ 𝑦 ∈ 𝑢) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑)))) → ([𝑦 / 𝑥]𝜑 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑)))
37	36	exp44 442	. . . . . . . . . 10 ⊢ ([𝑦 / 𝑥]𝜑 → (Tr 𝑢 → (𝑦 ∈ 𝑢 → (∀𝑧(𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑)) → ([𝑦 / 𝑥]𝜑 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑))))))
38	37	com3l 90	. . . . . . . . 9 ⊢ (Tr 𝑢 → (𝑦 ∈ 𝑢 → ([𝑦 / 𝑥]𝜑 → (∀𝑧(𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑)) → ([𝑦 / 𝑥]𝜑 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑))))))
39	38	imp4c 428	. . . . . . . 8 ⊢ (Tr 𝑢 → (((𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))) → ([𝑦 / 𝑥]𝜑 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑))))
40	39	eximdv 1945	. . . . . . 7 ⊢ (Tr 𝑢 → (∃𝑦((𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))) → ∃𝑦([𝑦 / 𝑥]𝜑 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑))))
41	40	ad2antlr 739	. . . . . 6 ⊢ (((𝑣 ∈ 𝑢 ∧ Tr 𝑢) ∧ [𝑣 / 𝑥]𝜑) → (∃𝑦((𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))) → ∃𝑦([𝑦 / 𝑥]𝜑 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑))))
42	26, 41	mpd 16	. . . . 5 ⊢ (((𝑣 ∈ 𝑢 ∧ Tr 𝑢) ∧ [𝑣 / 𝑥]𝜑) → ∃𝑦([𝑦 / 𝑥]𝜑 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑)))
43	42	ex 417	. . . 4 ⊢ ((𝑣 ∈ 𝑢 ∧ Tr 𝑢) → ([𝑣 / 𝑥]𝜑 → ∃𝑦([𝑦 / 𝑥]𝜑 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑))))
44		regsfromregtco.2	. . . . 5 ⊢ ∃𝑢(𝑣 ∈ 𝑢 ∧ ∀𝑡(𝑡 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑡 → 𝑠 ∈ 𝑢)))
45		dftr3 5222	. . . . . . . 8 ⊢ (Tr 𝑢 ↔ ∀𝑡 ∈ 𝑢 𝑡 ⊆ 𝑢)
46		df-ss 3921	. . . . . . . . 9 ⊢ (𝑡 ⊆ 𝑢 ↔ ∀𝑠(𝑠 ∈ 𝑡 → 𝑠 ∈ 𝑢))
47	46	ralbii 3109	. . . . . . . 8 ⊢ (∀𝑡 ∈ 𝑢 𝑡 ⊆ 𝑢 ↔ ∀𝑡 ∈ 𝑢 ∀𝑠(𝑠 ∈ 𝑡 → 𝑠 ∈ 𝑢))
48		df-ral 3078	. . . . . . . 8 ⊢ (∀𝑡 ∈ 𝑢 ∀𝑠(𝑠 ∈ 𝑡 → 𝑠 ∈ 𝑢) ↔ ∀𝑡(𝑡 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑡 → 𝑠 ∈ 𝑢)))
49	45, 47, 48	3bitri 300	. . . . . . 7 ⊢ (Tr 𝑢 ↔ ∀𝑡(𝑡 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑡 → 𝑠 ∈ 𝑢)))
50	49	anbi2i 634	. . . . . 6 ⊢ ((𝑣 ∈ 𝑢 ∧ Tr 𝑢) ↔ (𝑣 ∈ 𝑢 ∧ ∀𝑡(𝑡 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑡 → 𝑠 ∈ 𝑢))))
51	50	exbii 1876	. . . . 5 ⊢ (∃𝑢(𝑣 ∈ 𝑢 ∧ Tr 𝑢) ↔ ∃𝑢(𝑣 ∈ 𝑢 ∧ ∀𝑡(𝑡 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑡 → 𝑠 ∈ 𝑢))))
52	44, 51	mpbir 234	. . . 4 ⊢ ∃𝑢(𝑣 ∈ 𝑢 ∧ Tr 𝑢)
53	43, 52	exlimiiv 1959	. . 3 ⊢ ([𝑣 / 𝑥]𝜑 → ∃𝑦([𝑦 / 𝑥]𝜑 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑)))
54	53	exlimiv 1958	. 2 ⊢ (∃𝑣[𝑣 / 𝑥]𝜑 → ∃𝑦([𝑦 / 𝑥]𝜑 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑)))
55		nfv 1942	. . . 4 ⊢ Ⅎ𝑣𝜑
56	55	sb8ef 2385	. . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑣[𝑣 / 𝑥]𝜑)
57	56	bicomi 227	. 2 ⊢ (∃𝑣[𝑣 / 𝑥]𝜑 ↔ ∃𝑥𝜑)
58		sb6 2117	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
59		sb6 2117	. . . . . . 7 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑))
60	59	notbii 323	. . . . . 6 ⊢ (¬ [𝑧 / 𝑥]𝜑 ↔ ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))
61	60	imbi2i 339	. . . . 5 ⊢ ((𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑) ↔ (𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
62	61	albii 1847	. . . 4 ⊢ (∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
63	58, 62	anbi12i 639	. . 3 ⊢ (([𝑦 / 𝑥]𝜑 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑)) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
64	63	exbii 1876	. 2 ⊢ (∃𝑦([𝑦 / 𝑥]𝜑 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑)) ↔ ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
65	54, 57, 64	3imtr3i 294	1 ⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1566   = wceq 1568  ∃wex 1807  [wsb 2094   ∈ wcel 2141  ∀wral 3077  {crab 3414   ⊆ wss 3904  Tr wtr 5217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-ex 1808  df-nf 1812  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3415  df-v 3455  df-in 3911  df-ss 3921  df-pw 4563  df-uni 4872  df-tr 5218

This theorem is referenced by: (None)