Theorem ralcom2 3290

Description: Commutation of restricted universal quantifiers. Note that 𝑥 and 𝑦 need not be disjoint (this makes the proof longer). If 𝑥 and 𝑦 are disjoint, then one may use ralcom 3284. (Contributed by NM, 24-Nov-1994.) (Proof shortened by Mario Carneiro, 17-Oct-2016.)

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 2842	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	sps 2169	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
3	2	imbi1d 333	. . . . . . . . 9 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜑)))
4	3	dral1 2405	. . . . . . . 8 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)))
5	4	bicomd 215	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦(𝑦 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)))
6		df-ral 3095	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
7		df-ral 3095	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
8	5, 6, 7	3bitr4g 306	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))
9	2, 8	imbi12d 336	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜑) ↔ (𝑦 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑)))
10	9	dral1 2405	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑)))
11		df-ral 3095	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜑))
12		df-ral 3095	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))
13	10, 11, 12	3bitr4g 306	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑))
14	13	biimpd 221	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑))
15		nfnae 2400	. . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
16		nfra2 3128	. . . . 5 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑
17	15, 16	nfan 1946	. . . 4 ⊢ Ⅎ𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑)
18		nfnae 2400	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
19		nfra1 3123	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑
20	18, 19	nfan 1946	. . . . . . 7 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑)
21		nfcvf 2960	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
22	21	adantr 474	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) → Ⅎ𝑥𝑦)
23		nfcvd 2935	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) → Ⅎ𝑥𝐴)
24	22, 23	nfeld 2943	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) → Ⅎ𝑥 𝑦 ∈ 𝐴)
25	20, 24	nfan1 2183	. . . . . 6 ⊢ Ⅎ𝑥((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) ∧ 𝑦 ∈ 𝐴)
26		rsp2 3118	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝜑))
27	26	ancomsd 459	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝜑))
28	27	expdimp 446	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝐴 → 𝜑))
29	28	adantll 704	. . . . . 6 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝐴 → 𝜑))
30	25, 29	ralrimi 3139	. . . . 5 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) ∧ 𝑦 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 𝜑)
31	30	ex 403	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) → (𝑦 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))
32	17, 31	ralrimi 3139	. . 3 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) → ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑)
33	32	ex 403	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑))
34	14, 33	pm2.61i 177	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1599   ∈ wcel 2107  Ⅎwnfc 2919  ∀wral 3090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095

This theorem is referenced by:  tz7.48lem  7819  imo72b2  39431  tratrb  39696  tratrbVD  40030