Theorem mh-setindnd 36735

Description: A version of mh-setind 36734 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 2191	. . . . . 6 ⊢ (∀𝑦𝜑 → 𝜑)
2	1	imim2i 16	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 → ∀𝑦𝜑) → (𝑥 ∈ 𝑦 → 𝜑))
3	2	alimi 1813	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 ∈ 𝑦 → 𝜑))
4	3	imim1i 63	. . 3 ⊢ ((∀𝑥(𝑥 ∈ 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → (∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)))
5	4	alimi 1813	. 2 ⊢ (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → ∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)))
6		elirrv 9505	. . . . . . . 8 ⊢ ¬ 𝑥 ∈ 𝑥
7		elequ2 2129	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑥 ∈ 𝑦))
8	6, 7	mtbii 326	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦)
9	8	pm2.21d 121	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑦𝜑))
10	9	alimi 1813	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑))
11		sp 2191	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
12	1	a1i 11	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → 𝜑))
13	11, 12	embantd 59	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑦 → ∀𝑦𝜑) → 𝜑))
14	13	spsd 2195	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑) → 𝜑))
15	10, 14	embantd 59	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ((∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → 𝜑))
16	15	spsd 2195	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → 𝜑))
17		nfnae 2439	. . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
18		nfnae 2439	. . . . . . 7 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
19		dveel1 2466	. . . . . . . . . 10 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 ∈ 𝑧 → ∀𝑦 𝑥 ∈ 𝑧))
20	19	naecoms 2434	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → ∀𝑦 𝑥 ∈ 𝑧))
21	17, 20	nf5d 2291	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 ∈ 𝑧)
22		nfa1 2157	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦𝜑
23	22	a1i 11	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦∀𝑦𝜑)
24	21, 23	nfimd 1896	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(𝑥 ∈ 𝑧 → ∀𝑦𝜑))
25	18, 24	nfald 2334	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑))
26		nfeqf1 2384	. . . . . . . . 9 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥 = 𝑧)
27	26	naecoms 2434	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 = 𝑧)
28	27, 23	nfimd 1896	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(𝑥 = 𝑧 → ∀𝑦𝜑))
29	18, 28	nfald 2334	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑))
30	25, 29	nfimd 1896	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑)))
31		nfeqf2 2382	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
32	18, 31	nfan1 2208	. . . . . . . 8 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦)
33		elequ2 2129	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑦))
34	33	imbi1d 341	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑥 ∈ 𝑧 → ∀𝑦𝜑) ↔ (𝑥 ∈ 𝑦 → ∀𝑦𝜑)))
35	34	adantl 481	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → ((𝑥 ∈ 𝑧 → ∀𝑦𝜑) ↔ (𝑥 ∈ 𝑦 → ∀𝑦𝜑)))
36	32, 35	albid 2230	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → (∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑) ↔ ∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑)))
37		equequ2 2028	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
38	37	imbi1d 341	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑥 = 𝑧 → ∀𝑦𝜑) ↔ (𝑥 = 𝑦 → ∀𝑦𝜑)))
39	38	adantl 481	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → ((𝑥 = 𝑧 → ∀𝑦𝜑) ↔ (𝑥 = 𝑦 → ∀𝑦𝜑)))
40	32, 39	albid 2230	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → (∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)))
41	36, 40	imbi12d 344	. . . . . 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 412	. . . . 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 2343	. . . 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 36734	. . . . 5 ⊢ (∀𝑧(∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑)) → ∀𝑦𝜑)
45	44	19.21bi 2197	. . . 4 ⊢ (∀𝑧(∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑)) → 𝜑)
46	43, 45	biimtrrdi 254	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → 𝜑))
47	16, 46	pm2.61i 182	. 2 ⊢ (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → 𝜑)
48	5, 47	syl 17	1 ⊢ (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-13 2377  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682  ax-reg 9500  ax-inf2 9553

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342

This theorem is referenced by: (None)