Theorem mh-setindnd 37164

Description: A version of mh-setind 37163 with no distinct variable conditions. (Contributed by Matthew House, 5-Mar-2026.) (New usage is discouraged.)

Assertion

Ref	Expression
mh-setindnd	⊢ (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → 𝜑)

Proof of Theorem mh-setindnd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sp 2221	. . . . . 6 ⊢ (∀𝑦𝜑 → 𝜑)
2	1	imim2i 17	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 → ∀𝑦𝜑) → (𝑥 ∈ 𝑦 → 𝜑))
3	2	alimi 1844	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 ∈ 𝑦 → 𝜑))
4	3	imim1i 64	. . 3 ⊢ ((∀𝑥(𝑥 ∈ 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → (∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)))
5	4	alimi 1844	. 2 ⊢ (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → ∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)))
6		elirrv 9573	. . . . . . . 8 ⊢ ¬ 𝑥 ∈ 𝑥
7		elequ2 2160	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑥 ∈ 𝑦))
8	6, 7	mtbii 329	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦)
9	8	pm2.21d 122	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑦𝜑))
10	9	alimi 1844	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑))
11		sp 2221	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
12	1	a1i 11	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → 𝜑))
13	11, 12	embantd 60	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑦 → ∀𝑦𝜑) → 𝜑))
14	13	spsd 2225	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑) → 𝜑))
15	10, 14	embantd 60	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ((∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → 𝜑))
16	15	spsd 2225	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → 𝜑))
17		nfnae 2465	. . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
18		nfnae 2465	. . . . . . 7 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
19		dveel1 2492	. . . . . . . . . 10 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 ∈ 𝑧 → ∀𝑦 𝑥 ∈ 𝑧))
20	19	naecoms 2460	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → ∀𝑦 𝑥 ∈ 𝑧))
21	17, 20	nf5d 2319	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 ∈ 𝑧)
22		nfa1 2188	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦𝜑
23	22	a1i 11	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦∀𝑦𝜑)
24	21, 23	nfimd 1927	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(𝑥 ∈ 𝑧 → ∀𝑦𝜑))
25	18, 24	nfald 2360	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑))
26		nfeqf1 2410	. . . . . . . . 9 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥 = 𝑧)
27	26	naecoms 2460	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 = 𝑧)
28	27, 23	nfimd 1927	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(𝑥 = 𝑧 → ∀𝑦𝜑))
29	18, 28	nfald 2360	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑))
30	25, 29	nfimd 1927	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑)))
31		nfeqf2 2408	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
32	18, 31	nfan1 2238	. . . . . . . 8 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦)
33		elequ2 2160	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑦))
34	33	imbi1d 344	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑥 ∈ 𝑧 → ∀𝑦𝜑) ↔ (𝑥 ∈ 𝑦 → ∀𝑦𝜑)))
35	34	adantl 487	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → ((𝑥 ∈ 𝑧 → ∀𝑦𝜑) ↔ (𝑥 ∈ 𝑦 → ∀𝑦𝜑)))
36	32, 35	albid 2260	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → (∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑) ↔ ∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑)))
37		equequ2 2059	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
38	37	imbi1d 344	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑥 = 𝑧 → ∀𝑦𝜑) ↔ (𝑥 = 𝑦 → ∀𝑦𝜑)))
39	38	adantl 487	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → ((𝑥 = 𝑧 → ∀𝑦𝜑) ↔ (𝑥 = 𝑦 → ∀𝑦𝜑)))
40	32, 39	albid 2260	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → (∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)))
41	36, 40	imbi12d 347	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → ((∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑)) ↔ (∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑))))
42	41	ex 418	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ((∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑)) ↔ (∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)))))
43	17, 30, 42	cbvaldw 2369	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑧(∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑)) ↔ ∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑))))
44		mh-setind 37163	. . . . 5 ⊢ (∀𝑧(∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑)) → ∀𝑦𝜑)
45	44	19.21bi 2227	. . . 4 ⊢ (∀𝑧(∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑)) → 𝜑)
46	43, 45	biimtrrdi 257	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → 𝜑))
47	16, 46	pm2.61i 184	. 2 ⊢ (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → 𝜑)
48	5, 47	syl 18	1 ⊢ (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-13 2403  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740  ax-reg 9568  ax-inf2 9624

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-om 7867  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403

This theorem is used by: (None)