Theorem dfid3 5523

Description: A stronger version of df-id 5520 that does not require 𝑥 and 𝑦 to be disjoint. This is not the definition since, in order to pass our definition soundness test, a definition has to have disjoint dummy variables, see conventions 30495. The proof can be instructive in showing how disjoint variable conditions may be eliminated, a task that is not necessarily obvious. (Contributed by NM, 5-Feb-2008.) (Revised by Mario Carneiro, 18-Nov-2016.)

Use df-id 5520 instead to make the semantics of the constructor df-opab 5142 clearer (in usages, 𝑥, 𝑦 will typically be dummy variables, so can be assumed disjoint). (New usage is discouraged.)

Assertion

Ref	Expression
dfid3	⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}

Proof of Theorem dfid3

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-id 5520	. 2 ⊢ I = {⟨𝑥, 𝑧⟩ ∣ 𝑥 = 𝑧}
2		equcom 2025	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
3	2	anbi1ci 632	. . . . . . . . . 10 ⊢ ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑧 = 𝑥 ∧ 𝑤 = ⟨𝑥, 𝑧⟩))
4	3	exbii 1855	. . . . . . . . 9 ⊢ (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑤 = ⟨𝑥, 𝑧⟩))
5		opeq2 4812	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
6	5	eqeq2d 2751	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑤 = ⟨𝑥, 𝑧⟩ ↔ 𝑤 = ⟨𝑥, 𝑥⟩))
7	6	equsexvw 2012	. . . . . . . . 9 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑤 = ⟨𝑥, 𝑧⟩) ↔ 𝑤 = ⟨𝑥, 𝑥⟩)
8		equid 2019	. . . . . . . . . 10 ⊢ 𝑥 = 𝑥
9	8	biantru 534	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑥, 𝑥⟩ ↔ (𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
10	4, 7, 9	3bitri 298	. . . . . . . 8 ⊢ (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
11	10	exbii 1855	. . . . . . 7 ⊢ (∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
12		nfe1 2161	. . . . . . . 8 ⊢ Ⅎ𝑥∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥)
13	12	19.9 2217	. . . . . . 7 ⊢ (∃𝑥∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ ∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
14	11, 13	bitr4i 279	. . . . . 6 ⊢ (∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
15		opeq2 4812	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
16	15	eqeq2d 2751	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑤 = ⟨𝑥, 𝑥⟩ ↔ 𝑤 = ⟨𝑥, 𝑦⟩))
17		equequ2 2033	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑥 ↔ 𝑥 = 𝑦))
18	16, 17	anbi12d 638	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
19	18	sps 2197	. . . . . . . 8 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
20	19	drex1 2449	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
21	20	drex2 2450	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
22	14, 21	bitrid 284	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
23		nfnae 2442	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
24		nfnae 2442	. . . . . . 7 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
25		nfcvd 2903	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑤)
26		nfcvf2 2929	. . . . . . . . . 10 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)
27		nfcvd 2903	. . . . . . . . . 10 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑧)
28	26, 27	nfopd 4828	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦⟨𝑥, 𝑧⟩)
29	25, 28	nfeqd 2912	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑤 = ⟨𝑥, 𝑧⟩)
30	26, 27	nfeqd 2912	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 = 𝑧)
31	29, 30	nfand 1904	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧))
32		opeq2 4812	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑦⟩)
33	32	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑤 = ⟨𝑥, 𝑧⟩ ↔ 𝑤 = ⟨𝑥, 𝑦⟩))
34		equequ2 2033	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
35	33, 34	anbi12d 638	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
36	35	a1i 11	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦))))
37	24, 31, 36	cbvexd 2416	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
38	23, 37	exbid 2235	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
39	22, 38	pm2.61i 183	. . . 4 ⊢ (∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦))
40	39	abbii 2807	. . 3 ⊢ {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧)} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)}
41		df-opab 5142	. . 3 ⊢ {⟨𝑥, 𝑧⟩ ∣ 𝑥 = 𝑧} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧)}
42		df-opab 5142	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)}
43	40, 41, 42	3eqtr4i 2773	. 2 ⊢ {⟨𝑥, 𝑧⟩ ∣ 𝑥 = 𝑧} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
44	1, 43	eqtri 2763	1 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547  ∃wex 1786  {cab 2718  ⟨cop 4568  {copab 5141   I cid 5519

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

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-opab 5142  df-id 5520

This theorem is referenced by: (None)