Theorem mh-setindnd 37295

Description: A version of mh-setind 37294 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 2220	. . . . . 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 9575	. . . . . . . 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 2220	. . . . . . 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 2224	. . . . 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 2224	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → 𝜑))
17		nfnae 2464	. . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
18		nfnae 2464	. . . . . . 7 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
19		dveel1 2491	. . . . . . . . . 10 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 ∈ 𝑧 → ∀𝑦 𝑥 ∈ 𝑧))
20	19	naecoms 2459	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → ∀𝑦 𝑥 ∈ 𝑧))
21	17, 20	nf5d 2318	. . . . . . . 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 2359	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑))
26		nfeqf1 2409	. . . . . . . . 9 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥 = 𝑧)
27	26	naecoms 2459	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 = 𝑧)
28	27, 23	nfimd 1927	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(𝑥 = 𝑧 → ∀𝑦𝜑))
29	18, 28	nfald 2359	. . . . . 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 2407	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
32	18, 31	nfan1 2237	. . . . . . . 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 2259	. . . . . . 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 2259	. . . . . . 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 2368	. . . 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 37294	. . . . 5 ⊢ (∀𝑧(∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑)) → ∀𝑦𝜑)
45	44	19.21bi 2226	. . . 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 2213  ax-13 2402  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7740  ax-reg 9570  ax-inf2 9626

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-om 7867  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402

This theorem is used by: (None)