Theorem regsfromregtco 36903

Description: Derivation of ax-regs 35426 from ax-reg 9542 + ax-tco 36837. (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 3460	. . . . . . . . 9 ⊢ 𝑣 ∈ V
2		elequ1 2151	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (𝑦 ∈ 𝑢 ↔ 𝑣 ∈ 𝑢))
3		sbequ 2118	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → ([𝑦 / 𝑥]𝜑 ↔ [𝑣 / 𝑥]𝜑))
4	2, 3	anbi12d 641	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → ((𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝑣 ∈ 𝑢 ∧ [𝑣 / 𝑥]𝜑)))
5	1, 4	spcev 3567	. . . . . . . 8 ⊢ ((𝑣 ∈ 𝑢 ∧ [𝑣 / 𝑥]𝜑) → ∃𝑦(𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑))
6	5	adantlr 725	. . . . . . 7 ⊢ (((𝑣 ∈ 𝑢 ∧ Tr 𝑢) ∧ [𝑣 / 𝑥]𝜑) → ∃𝑦(𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑))
7		vex 3460	. . . . . . . . 9 ⊢ 𝑢 ∈ V
8	7	rabex 5297	. . . . . . . 8 ⊢ {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} ∈ V
9		eleq2 2853	. . . . . . . . . . 11 ⊢ (𝑤 = {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑}))
10		sbequ 2118	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑦 → ([𝑟 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
11	10	elrab 3652	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} ↔ (𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑))
12	9, 11	bitrdi 289	. . . . . . . . . 10 ⊢ (𝑤 = {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} → (𝑦 ∈ 𝑤 ↔ (𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑)))
13	12	exbidv 1943	. . . . . . . . 9 ⊢ (𝑤 = {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} → (∃𝑦 𝑦 ∈ 𝑤 ↔ ∃𝑦(𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑)))
14		eleq2 2853	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑}))
15		sbequ 2118	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑧 → ([𝑟 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
16	15	elrab 3652	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} ↔ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))
17	14, 16	bitrdi 289	. . . . . . . . . . . . . 14 ⊢ (𝑤 = {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} → (𝑧 ∈ 𝑤 ↔ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑)))
18	17	notbid 320	. . . . . . . . . . . . 13 ⊢ (𝑤 = {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} → (¬ 𝑧 ∈ 𝑤 ↔ ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑)))
19	18	imbi2d 342	. . . . . . . . . . . 12 ⊢ (𝑤 = {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} → ((𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑤) ↔ (𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))))
20	19	albidv 1942	. . . . . . . . . . 11 ⊢ (𝑤 = {𝑟 ∈ 𝑢 ∣ [𝑟 / 𝑥]𝜑} → (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑤) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))))
21	12, 20	anbi12d 641	. . . . . . . . . 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 1943	. . . . . . . . 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 346	. . . . . . . 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 3527	. . . . . . 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 17	. . . . . 6 ⊢ (((𝑣 ∈ 𝑢 ∧ Tr 𝑢) ∧ [𝑣 / 𝑥]𝜑) → ∃𝑦((𝑦 ∈ 𝑢 ∧ [𝑦 / 𝑥]𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))))
27		imnan 403	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑢 → ¬ [𝑧 / 𝑥]𝜑) ↔ ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))
28		trel 5217	. . . . . . . . . . . . . . . . . 18 ⊢ (Tr 𝑢 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑢) → 𝑧 ∈ 𝑢))
29	28	imp 410	. . . . . . . . . . . . . . . . 17 ⊢ ((Tr 𝑢 ∧ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑢)) → 𝑧 ∈ 𝑢)
30	29	anass1rs 665	. . . . . . . . . . . . . . . 16 ⊢ (((Tr 𝑢 ∧ 𝑦 ∈ 𝑢) ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ 𝑢)
31		imbibi 394	. . . . . . . . . . . . . . . 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 68	. . . . . . . . . . . . . . 15 ⊢ (((Tr 𝑢 ∧ 𝑦 ∈ 𝑢) ∧ 𝑧 ∈ 𝑦) → (¬ [𝑧 / 𝑥]𝜑 ↔ ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑)))
33	32	pm5.74da 813	. . . . . . . . . . . . . 14 ⊢ ((Tr 𝑢 ∧ 𝑦 ∈ 𝑢) → ((𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑) ↔ (𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))))
34	33	albidv 1942	. . . . . . . . . . . . 13 ⊢ ((Tr 𝑢 ∧ 𝑦 ∈ 𝑢) → (∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))))
35	34	biimpar 481	. . . . . . . . . . . 12 ⊢ (((Tr 𝑢 ∧ 𝑦 ∈ 𝑢) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ (𝑧 ∈ 𝑢 ∧ [𝑧 / 𝑥]𝜑))) → ∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑))
36	35	anim2i 626	. . . . . . . . . . 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 441	. . . . . . . . . 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 89	. . . . . . . . 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 427	. . . . . . . 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 1939	. . . . . . 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 737	. . . . . 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 15	. . . . 5 ⊢ (((𝑣 ∈ 𝑢 ∧ Tr 𝑢) ∧ [𝑣 / 𝑥]𝜑) → ∃𝑦([𝑦 / 𝑥]𝜑 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑)))
43	42	ex 416	. . . 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 5214	. . . . . . . 8 ⊢ (Tr 𝑢 ↔ ∀𝑡 ∈ 𝑢 𝑡 ⊆ 𝑢)
46		df-ss 3923	. . . . . . . . 9 ⊢ (𝑡 ⊆ 𝑢 ↔ ∀𝑠(𝑠 ∈ 𝑡 → 𝑠 ∈ 𝑢))
47	46	ralbii 3110	. . . . . . . 8 ⊢ (∀𝑡 ∈ 𝑢 𝑡 ⊆ 𝑢 ↔ ∀𝑡 ∈ 𝑢 ∀𝑠(𝑠 ∈ 𝑡 → 𝑠 ∈ 𝑢))
48		df-ral 3079	. . . . . . . 8 ⊢ (∀𝑡 ∈ 𝑢 ∀𝑠(𝑠 ∈ 𝑡 → 𝑠 ∈ 𝑢) ↔ ∀𝑡(𝑡 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑡 → 𝑠 ∈ 𝑢)))
49	45, 47, 48	3bitri 299	. . . . . . 7 ⊢ (Tr 𝑢 ↔ ∀𝑡(𝑡 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑡 → 𝑠 ∈ 𝑢)))
50	49	anbi2i 632	. . . . . 6 ⊢ ((𝑣 ∈ 𝑢 ∧ Tr 𝑢) ↔ (𝑣 ∈ 𝑢 ∧ ∀𝑡(𝑡 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑡 → 𝑠 ∈ 𝑢))))
51	50	exbii 1870	. . . . 5 ⊢ (∃𝑢(𝑣 ∈ 𝑢 ∧ Tr 𝑢) ↔ ∃𝑢(𝑣 ∈ 𝑢 ∧ ∀𝑡(𝑡 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑡 → 𝑠 ∈ 𝑢))))
52	44, 51	mpbir 233	. . . 4 ⊢ ∃𝑢(𝑣 ∈ 𝑢 ∧ Tr 𝑢)
53	43, 52	exlimiiv 1953	. . 3 ⊢ ([𝑣 / 𝑥]𝜑 → ∃𝑦([𝑦 / 𝑥]𝜑 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑)))
54	53	exlimiv 1952	. 2 ⊢ (∃𝑣[𝑣 / 𝑥]𝜑 → ∃𝑦([𝑦 / 𝑥]𝜑 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑)))
55		nfv 1936	. . . 4 ⊢ Ⅎ𝑣𝜑
56	55	sb8ef 2388	. . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑣[𝑣 / 𝑥]𝜑)
57	56	bicomi 226	. 2 ⊢ (∃𝑣[𝑣 / 𝑥]𝜑 ↔ ∃𝑥𝜑)
58		sb6 2120	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
59		sb6 2120	. . . . . . 7 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑))
60	59	notbii 322	. . . . . 6 ⊢ (¬ [𝑧 / 𝑥]𝜑 ↔ ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))
61	60	imbi2i 338	. . . . 5 ⊢ ((𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑) ↔ (𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
62	61	albii 1841	. . . 4 ⊢ (∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
63	58, 62	anbi12i 637	. . 3 ⊢ (([𝑦 / 𝑥]𝜑 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ [𝑧 / 𝑥]𝜑)) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
64	63	exbii 1870	. 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 293	1 ⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1560   = wceq 1562  ∃wex 1801  [wsb 2092   ∈ wcel 2144  ∀wral 3078  {crab 3416   ⊆ wss 3906  Tr wtr 5209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rab 3417  df-v 3458  df-in 3913  df-ss 3923  df-pw 4559  df-uni 4868  df-tr 5210

This theorem is referenced by: (None)