Theorem mh-setindnd 37417

Description: A version of mh-setind 37416 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 2219	. . . . . 6 ⊢ (∀𝑦𝜑 → 𝜑)
2	1	imim2i 17	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 → ∀𝑦𝜑) → (𝑥 ∈ 𝑦 → 𝜑))
3	2	alimi 1843	. . . 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 1843	. 2 ⊢ (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → ∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)))
6		elirrv 9594	. . . . . . . 8 ⊢ ¬ 𝑥 ∈ 𝑥
7		elequ2 2159	. . . . . . . 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 1843	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑))
11		sp 2219	. . . . . . 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 2223	. . . . 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 2223	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → 𝜑))
17		nfnae 2463	. . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
18		nfnae 2463	. . . . . . 7 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
19		dveel1 2490	. . . . . . . . . 10 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 ∈ 𝑧 → ∀𝑦 𝑥 ∈ 𝑧))
20	19	naecoms 2458	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → ∀𝑦 𝑥 ∈ 𝑧))
21	17, 20	nf5d 2317	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 ∈ 𝑧)
22		nfa1 2187	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦𝜑
23	22	a1i 11	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦∀𝑦𝜑)
24	21, 23	nfimd 1926	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(𝑥 ∈ 𝑧 → ∀𝑦𝜑))
25	18, 24	nfald 2358	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑))
26		nfeqf1 2408	. . . . . . . . 9 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥 = 𝑧)
27	26	naecoms 2458	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 = 𝑧)
28	27, 23	nfimd 1926	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(𝑥 = 𝑧 → ∀𝑦𝜑))
29	18, 28	nfald 2358	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑))
30	25, 29	nfimd 1926	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑)))
31		nfeqf2 2406	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
32	18, 31	nfan1 2236	. . . . . . . 8 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦)
33		elequ2 2159	. . . . . . . . . 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 2258	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → (∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑) ↔ ∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑)))
37		equequ2 2058	. . . . . . . . . 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 2258	. . . . . . 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 2367	. . . 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 37416	. . . . 5 ⊢ (∀𝑧(∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑)) → ∀𝑦𝜑)
45	44	19.21bi 2225	. . . 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 1567  Ⅎwnf 1815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-13 2401  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751  ax-reg 9589  ax-inf2 9645

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421

This theorem is used by: (None)