Theorem ralcom2 3265

Description: Commutation of restricted universal quantifiers. Note that 𝑥 and 𝑦 need not be disjoint (this makes the proof longer). This theorem relies on the full set of axioms up to ax-ext 2708 and it should no longer be used. Usage of ralcom 3257 is highly encouraged. (Contributed by NM, 24-Nov-1994.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (New usage is discouraged.)

Assertion

Ref	Expression
ralcom2	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑦,𝐴   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ralcom2

Step	Hyp	Ref	Expression
1		eleq1w 2813	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	sps 2184	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
3	2	imbi1d 345	. . . . . . . . 9 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜑)))
4	3	dral1 2438	. . . . . . . 8 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)))
5	4	bicomd 226	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦(𝑦 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)))
6		df-ral 3056	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
7		df-ral 3056	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
8	5, 6, 7	3bitr4g 317	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))
9	2, 8	imbi12d 348	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜑) ↔ (𝑦 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑)))
10	9	dral1 2438	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑)))
11		df-ral 3056	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜑))
12		df-ral 3056	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))
13	10, 11, 12	3bitr4g 317	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑))
14	13	biimpd 232	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑))
15		nfnae 2433	. . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
16		nfra2 3141	. . . . 5 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑
17	15, 16	nfan 1907	. . . 4 ⊢ Ⅎ𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑)
18		nfnae 2433	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
19		nfra1 3130	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑
20	18, 19	nfan 1907	. . . . . . 7 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑)
21		nfcvf 2926	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
22	21	adantr 484	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) → Ⅎ𝑥𝑦)
23		nfcvd 2898	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) → Ⅎ𝑥𝐴)
24	22, 23	nfeld 2908	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) → Ⅎ𝑥 𝑦 ∈ 𝐴)
25	20, 24	nfan1 2200	. . . . . 6 ⊢ Ⅎ𝑥((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) ∧ 𝑦 ∈ 𝐴)
26		rsp2 3124	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝜑))
27	26	ancomsd 469	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝜑))
28	27	expdimp 456	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝐴 → 𝜑))
29	28	adantll 714	. . . . . 6 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝐴 → 𝜑))
30	25, 29	ralrimi 3127	. . . . 5 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) ∧ 𝑦 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 𝜑)
31	30	ex 416	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) → (𝑦 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))
32	17, 31	ralrimi 3127	. . 3 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) → ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑)
33	32	ex 416	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑))
34	14, 33	pm2.61i 185	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541   ∈ wcel 2112  Ⅎwnfc 2877  ∀wral 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-13 2371  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ral 3056

This theorem is referenced by: (None)