Theorem dfid3 5186

Description: A stronger version of df-id 5185 that does not require 𝑥 and 𝑦 to be disjoint. This is not the "official" definition since our definition soundness check without this requirement would be much more complex. The proof can be instructive in showing how disjoint variable requirements may be eliminated, a task that is not necessarily obvious. (Contributed by NM, 5-Feb-2008.) (Revised by Mario Carneiro, 18-Nov-2016.) Use directly the definition df-id 5185 when sufficient, since the derivation of dfid3 5186 is nontrivial and uses auxiliary axioms ax-10 2183 to ax-13 2352. (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 5185	. 2 ⊢ I = {⟨𝑥, 𝑧⟩ ∣ 𝑥 = 𝑧}
2		ancom 452	. . . . . . . . . . 11 ⊢ ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑥 = 𝑧 ∧ 𝑤 = ⟨𝑥, 𝑧⟩))
3		equcom 2115	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
4	3	anbi1i 617	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑧 ∧ 𝑤 = ⟨𝑥, 𝑧⟩) ↔ (𝑧 = 𝑥 ∧ 𝑤 = ⟨𝑥, 𝑧⟩))
5	2, 4	bitri 266	. . . . . . . . . 10 ⊢ ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑧 = 𝑥 ∧ 𝑤 = ⟨𝑥, 𝑧⟩))
6	5	exbii 1943	. . . . . . . . 9 ⊢ (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑤 = ⟨𝑥, 𝑧⟩))
7		opeq2 4560	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
8	7	eqeq2d 2775	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑤 = ⟨𝑥, 𝑧⟩ ↔ 𝑤 = ⟨𝑥, 𝑥⟩))
9	8	equsexvw 2102	. . . . . . . . 9 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑤 = ⟨𝑥, 𝑧⟩) ↔ 𝑤 = ⟨𝑥, 𝑥⟩)
10		equid 2109	. . . . . . . . . 10 ⊢ 𝑥 = 𝑥
11	10	biantru 525	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑥, 𝑥⟩ ↔ (𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
12	6, 9, 11	3bitri 288	. . . . . . . 8 ⊢ (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
13	12	exbii 1943	. . . . . . 7 ⊢ (∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
14		nfe1 2192	. . . . . . . 8 ⊢ Ⅎ𝑥∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥)
15	14	19.9 2237	. . . . . . 7 ⊢ (∃𝑥∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ ∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
16	13, 15	bitr4i 269	. . . . . 6 ⊢ (∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
17		opeq2 4560	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
18	17	eqeq2d 2775	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑤 = ⟨𝑥, 𝑥⟩ ↔ 𝑤 = ⟨𝑥, 𝑦⟩))
19		equequ2 2123	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑥 ↔ 𝑥 = 𝑦))
20	18, 19	anbi12d 624	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
21	20	sps 2217	. . . . . . . 8 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
22	21	drex1 2421	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
23	22	drex2 2422	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
24	16, 23	syl5bb 274	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
25		nfnae 2414	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
26		nfnae 2414	. . . . . . 7 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
27		nfcvd 2908	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑤)
28		nfcvf2 2932	. . . . . . . . . 10 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)
29		nfcvd 2908	. . . . . . . . . 10 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑧)
30	28, 29	nfopd 4576	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦⟨𝑥, 𝑧⟩)
31	27, 30	nfeqd 2915	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑤 = ⟨𝑥, 𝑧⟩)
32	28, 29	nfeqd 2915	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 = 𝑧)
33	31, 32	nfand 1996	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧))
34		opeq2 4560	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑦⟩)
35	34	eqeq2d 2775	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑤 = ⟨𝑥, 𝑧⟩ ↔ 𝑤 = ⟨𝑥, 𝑦⟩))
36		equequ2 2123	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
37	35, 36	anbi12d 624	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
38	37	a1i 11	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦))))
39	26, 33, 38	cbvexd 2381	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
40	25, 39	exbid 2256	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
41	24, 40	pm2.61i 176	. . . 4 ⊢ (∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦))
42	41	abbii 2882	. . 3 ⊢ {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧)} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)}
43		df-opab 4872	. . 3 ⊢ {⟨𝑥, 𝑧⟩ ∣ 𝑥 = 𝑧} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧)}
44		df-opab 4872	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)}
45	42, 43, 44	3eqtr4i 2797	. 2 ⊢ {⟨𝑥, 𝑧⟩ ∣ 𝑥 = 𝑧} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
46	1, 45	eqtri 2787	1 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 197   ∧ wa 384  ∀wal 1650   = wceq 1652  ∃wex 1874  {cab 2751  ⟨cop 4340  {copab 4871   I cid 5184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-opab 4872  df-id 5185

This theorem is referenced by:  dfid2  5188