Theorem mh-setindnd 36772

Description: A version of mh-setind 36771 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 2195	. . . . . 6 ⊢ (∀𝑦𝜑 → 𝜑)
2	1	imim2i 16	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 → ∀𝑦𝜑) → (𝑥 ∈ 𝑦 → 𝜑))
3	2	alimi 1818	. . . 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 1818	. 2 ⊢ (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → ∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)))
6		elirrv 9509	. . . . . . . 8 ⊢ ¬ 𝑥 ∈ 𝑥
7		elequ2 2134	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑥 ∈ 𝑦))
8	6, 7	mtbii 327	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦)
9	8	pm2.21d 121	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑦𝜑))
10	9	alimi 1818	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑))
11		sp 2195	. . . . . . 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 2199	. . . . 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 2199	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → 𝜑))
17		nfnae 2442	. . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
18		nfnae 2442	. . . . . . 7 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
19		dveel1 2469	. . . . . . . . . 10 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 ∈ 𝑧 → ∀𝑦 𝑥 ∈ 𝑧))
20	19	naecoms 2437	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → ∀𝑦 𝑥 ∈ 𝑧))
21	17, 20	nf5d 2295	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 ∈ 𝑧)
22		nfa1 2162	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦𝜑
23	22	a1i 11	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦∀𝑦𝜑)
24	21, 23	nfimd 1901	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(𝑥 ∈ 𝑧 → ∀𝑦𝜑))
25	18, 24	nfald 2337	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑))
26		nfeqf1 2387	. . . . . . . . 9 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥 = 𝑧)
27	26	naecoms 2437	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 = 𝑧)
28	27, 23	nfimd 1901	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(𝑥 = 𝑧 → ∀𝑦𝜑))
29	18, 28	nfald 2337	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑))
30	25, 29	nfimd 1901	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑)))
31		nfeqf2 2385	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
32	18, 31	nfan1 2212	. . . . . . . 8 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦)
33		elequ2 2134	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑦))
34	33	imbi1d 342	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑥 ∈ 𝑧 → ∀𝑦𝜑) ↔ (𝑥 ∈ 𝑦 → ∀𝑦𝜑)))
35	34	adantl 482	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → ((𝑥 ∈ 𝑧 → ∀𝑦𝜑) ↔ (𝑥 ∈ 𝑦 → ∀𝑦𝜑)))
36	32, 35	albid 2234	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → (∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑) ↔ ∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑)))
37		equequ2 2033	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
38	37	imbi1d 342	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑥 = 𝑧 → ∀𝑦𝜑) ↔ (𝑥 = 𝑦 → ∀𝑦𝜑)))
39	38	adantl 482	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → ((𝑥 = 𝑧 → ∀𝑦𝜑) ↔ (𝑥 = 𝑦 → ∀𝑦𝜑)))
40	32, 39	albid 2234	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → (∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)))
41	36, 40	imbi12d 345	. . . . . 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 413	. . . . 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 2346	. . . 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 36771	. . . . 5 ⊢ (∀𝑧(∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑)) → ∀𝑦𝜑)
45	44	19.21bi 2201	. . . 4 ⊢ (∀𝑧(∀𝑥(𝑥 ∈ 𝑧 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑧 → ∀𝑦𝜑)) → 𝜑)
46	43, 45	biimtrrdi 255	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → 𝜑))
47	16, 46	pm2.61i 183	. 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 207   ∧ wa 396  ∀wal 1545  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-13 2380  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685  ax-reg 9504  ax-inf2 9560

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  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 7366  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346

This theorem is referenced by: (None)