Theorem dfid3 5585

Description: A stronger version of df-id 5582 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 30428. 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 5582 instead to make the semantics of the constructor df-opab 5210 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 5582	. 2 ⊢ I = {⟨𝑥, 𝑧⟩ ∣ 𝑥 = 𝑧}
2		equcom 2014	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
3	2	anbi1ci 626	. . . . . . . . . 10 ⊢ ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑧 = 𝑥 ∧ 𝑤 = ⟨𝑥, 𝑧⟩))
4	3	exbii 1844	. . . . . . . . 9 ⊢ (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑤 = ⟨𝑥, 𝑧⟩))
5		opeq2 4878	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
6	5	eqeq2d 2745	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑤 = ⟨𝑥, 𝑧⟩ ↔ 𝑤 = ⟨𝑥, 𝑥⟩))
7	6	equsexvw 2001	. . . . . . . . 9 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑤 = ⟨𝑥, 𝑧⟩) ↔ 𝑤 = ⟨𝑥, 𝑥⟩)
8		equid 2008	. . . . . . . . . 10 ⊢ 𝑥 = 𝑥
9	8	biantru 529	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑥, 𝑥⟩ ↔ (𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
10	4, 7, 9	3bitri 297	. . . . . . . 8 ⊢ (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
11	10	exbii 1844	. . . . . . 7 ⊢ (∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
12		nfe1 2147	. . . . . . . 8 ⊢ Ⅎ𝑥∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥)
13	12	19.9 2202	. . . . . . 7 ⊢ (∃𝑥∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ ∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
14	11, 13	bitr4i 278	. . . . . 6 ⊢ (∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
15		opeq2 4878	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
16	15	eqeq2d 2745	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑤 = ⟨𝑥, 𝑥⟩ ↔ 𝑤 = ⟨𝑥, 𝑦⟩))
17		equequ2 2022	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑥 ↔ 𝑥 = 𝑦))
18	16, 17	anbi12d 632	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
19	18	sps 2182	. . . . . . . 8 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
20	19	drex1 2443	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
21	20	drex2 2444	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
22	14, 21	bitrid 283	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
23		nfnae 2436	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
24		nfnae 2436	. . . . . . 7 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
25		nfcvd 2903	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑤)
26		nfcvf2 2930	. . . . . . . . . 10 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)
27		nfcvd 2903	. . . . . . . . . 10 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑧)
28	26, 27	nfopd 4894	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦⟨𝑥, 𝑧⟩)
29	25, 28	nfeqd 2913	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑤 = ⟨𝑥, 𝑧⟩)
30	26, 27	nfeqd 2913	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 = 𝑧)
31	29, 30	nfand 1894	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧))
32		opeq2 4878	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑦⟩)
33	32	eqeq2d 2745	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑤 = ⟨𝑥, 𝑧⟩ ↔ 𝑤 = ⟨𝑥, 𝑦⟩))
34		equequ2 2022	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
35	33, 34	anbi12d 632	. . . . . . . 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 2410	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
38	23, 37	exbid 2220	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
39	22, 38	pm2.61i 182	. . . 4 ⊢ (∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦))
40	39	abbii 2806	. . 3 ⊢ {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧)} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)}
41		df-opab 5210	. . 3 ⊢ {⟨𝑥, 𝑧⟩ ∣ 𝑥 = 𝑧} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧)}
42		df-opab 5210	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)}
43	40, 41, 42	3eqtr4i 2772	. 2 ⊢ {⟨𝑥, 𝑧⟩ ∣ 𝑥 = 𝑧} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
44	1, 43	eqtri 2762	1 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1534   = wceq 1536  ∃wex 1775  {cab 2711  ⟨cop 4636  {copab 5209   I cid 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-13 2374  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-opab 5210  df-id 5582

This theorem is referenced by:  dfid2OLD  5586