Theorem ralcom2 3104

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 3098. (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		eleq1 2689	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	sps 2055	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
3	2	imbi1d 331	. . . . . . . . 9 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜑)))
4	3	dral1 2325	. . . . . . . 8 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)))
5	4	bicomd 213	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦(𝑦 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)))
6		df-ral 2917	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
7		df-ral 2917	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
8	5, 6, 7	3bitr4g 303	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))
9	2, 8	imbi12d 334	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜑) ↔ (𝑦 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑)))
10	9	dral1 2325	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑)))
11		df-ral 2917	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜑))
12		df-ral 2917	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))
13	10, 11, 12	3bitr4g 303	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑))
14	13	biimpd 219	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑))
15		nfnae 2318	. . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
16		nfra2 2946	. . . . 5 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑
17	15, 16	nfan 1828	. . . 4 ⊢ Ⅎ𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑)
18		nfnae 2318	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
19		nfra1 2941	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑
20	18, 19	nfan 1828	. . . . . . 7 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑)
21		nfcvf 2788	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
22	21	adantr 481	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) → Ⅎ𝑥𝑦)
23		nfcvd 2765	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) → Ⅎ𝑥𝐴)
24	22, 23	nfeld 2773	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) → Ⅎ𝑥 𝑦 ∈ 𝐴)
25	20, 24	nfan1 2068	. . . . . 6 ⊢ Ⅎ𝑥((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) ∧ 𝑦 ∈ 𝐴)
26		rsp2 2936	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝜑))
27	26	ancomsd 470	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝜑))
28	27	expdimp 453	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝐴 → 𝜑))
29	28	adantll 750	. . . . . 6 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝐴 → 𝜑))
30	25, 29	ralrimi 2957	. . . . 5 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) ∧ 𝑦 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 𝜑)
31	30	ex 450	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) → (𝑦 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))
32	17, 31	ralrimi 2957	. . 3 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) → ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑)
33	32	ex 450	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑))
34	14, 33	pm2.61i 176	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   ∈ wcel 1990  Ⅎwnfc 2751  ∀wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917

This theorem is referenced by:  tz7.48lem  7536  imo72b2  38475  tratrb  38746  tratrbVD  39097