Theorem dfid3 4769

Description: A stronger version of df-id 4768 that doesn't require x and y to be distinct. Ordinarily, we wouldn't use this as a definition, since non-distinct dummy variables would make soundness verification more difficult (as the proof here shows). The proof can be instructive in showing how distinct 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.)

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 4768	. 2 ⊢ I = {⟨x, z⟩ ∣ x = z}
2		ancom 437	. . . . . . . . . . 11 ⊢ ((w = ⟨x, z⟩ ∧ x = z) ↔ (x = z ∧ w = ⟨x, z⟩))
3		equcom 1680	. . . . . . . . . . . 12 ⊢ (x = z ↔ z = x)
4	3	anbi1i 676	. . . . . . . . . . 11 ⊢ ((x = z ∧ w = ⟨x, z⟩) ↔ (z = x ∧ w = ⟨x, z⟩))
5	2, 4	bitri 240	. . . . . . . . . 10 ⊢ ((w = ⟨x, z⟩ ∧ x = z) ↔ (z = x ∧ w = ⟨x, z⟩))
6	5	exbii 1582	. . . . . . . . 9 ⊢ (∃z(w = ⟨x, z⟩ ∧ x = z) ↔ ∃z(z = x ∧ w = ⟨x, z⟩))
7		vex 2863	. . . . . . . . . 10 ⊢ x ∈ V
8		opeq2 4580	. . . . . . . . . . 11 ⊢ (z = x → ⟨x, z⟩ = ⟨x, x⟩)
9	8	eqeq2d 2364	. . . . . . . . . 10 ⊢ (z = x → (w = ⟨x, z⟩ ↔ w = ⟨x, x⟩))
10	7, 9	ceqsexv 2895	. . . . . . . . 9 ⊢ (∃z(z = x ∧ w = ⟨x, z⟩) ↔ w = ⟨x, x⟩)
11		equid 1676	. . . . . . . . . 10 ⊢ x = x
12	11	biantru 491	. . . . . . . . 9 ⊢ (w = ⟨x, x⟩ ↔ (w = ⟨x, x⟩ ∧ x = x))
13	6, 10, 12	3bitri 262	. . . . . . . 8 ⊢ (∃z(w = ⟨x, z⟩ ∧ x = z) ↔ (w = ⟨x, x⟩ ∧ x = x))
14	13	exbii 1582	. . . . . . 7 ⊢ (∃x∃z(w = ⟨x, z⟩ ∧ x = z) ↔ ∃x(w = ⟨x, x⟩ ∧ x = x))
15		nfe1 1732	. . . . . . . 8 ⊢ Ⅎx∃x(w = ⟨x, x⟩ ∧ x = x)
16	15	19.9 1783	. . . . . . 7 ⊢ (∃x∃x(w = ⟨x, x⟩ ∧ x = x) ↔ ∃x(w = ⟨x, x⟩ ∧ x = x))
17	14, 16	bitr4i 243	. . . . . 6 ⊢ (∃x∃z(w = ⟨x, z⟩ ∧ x = z) ↔ ∃x∃x(w = ⟨x, x⟩ ∧ x = x))
18		opeq2 4580	. . . . . . . . . . 11 ⊢ (x = y → ⟨x, x⟩ = ⟨x, y⟩)
19	18	eqeq2d 2364	. . . . . . . . . 10 ⊢ (x = y → (w = ⟨x, x⟩ ↔ w = ⟨x, y⟩))
20		equequ2 1686	. . . . . . . . . 10 ⊢ (x = y → (x = x ↔ x = y))
21	19, 20	anbi12d 691	. . . . . . . . 9 ⊢ (x = y → ((w = ⟨x, x⟩ ∧ x = x) ↔ (w = ⟨x, y⟩ ∧ x = y)))
22	21	sps 1754	. . . . . . . 8 ⊢ (∀x x = y → ((w = ⟨x, x⟩ ∧ x = x) ↔ (w = ⟨x, y⟩ ∧ x = y)))
23	22	drex1 1967	. . . . . . 7 ⊢ (∀x x = y → (∃x(w = ⟨x, x⟩ ∧ x = x) ↔ ∃y(w = ⟨x, y⟩ ∧ x = y)))
24	23	drex2 1968	. . . . . 6 ⊢ (∀x x = y → (∃x∃x(w = ⟨x, x⟩ ∧ x = x) ↔ ∃x∃y(w = ⟨x, y⟩ ∧ x = y)))
25	17, 24	syl5bb 248	. . . . 5 ⊢ (∀x x = y → (∃x∃z(w = ⟨x, z⟩ ∧ x = z) ↔ ∃x∃y(w = ⟨x, y⟩ ∧ x = y)))
26		nfnae 1956	. . . . . 6 ⊢ Ⅎx ¬ ∀x x = y
27		nfnae 1956	. . . . . . 7 ⊢ Ⅎy ¬ ∀x x = y
28		nfcvd 2491	. . . . . . . . 9 ⊢ (¬ ∀x x = y → Ⅎyw)
29		nfcvf2 2513	. . . . . . . . . 10 ⊢ (¬ ∀x x = y → Ⅎyx)
30		nfcvd 2491	. . . . . . . . . 10 ⊢ (¬ ∀x x = y → Ⅎyz)
31	29, 30	nfopd 4606	. . . . . . . . 9 ⊢ (¬ ∀x x = y → Ⅎy⟨x, z⟩)
32	28, 31	nfeqd 2504	. . . . . . . 8 ⊢ (¬ ∀x x = y → Ⅎy w = ⟨x, z⟩)
33	29, 30	nfeqd 2504	. . . . . . . 8 ⊢ (¬ ∀x x = y → Ⅎy x = z)
34	32, 33	nfand 1822	. . . . . . 7 ⊢ (¬ ∀x x = y → Ⅎy(w = ⟨x, z⟩ ∧ x = z))
35		opeq2 4580	. . . . . . . . . 10 ⊢ (z = y → ⟨x, z⟩ = ⟨x, y⟩)
36	35	eqeq2d 2364	. . . . . . . . 9 ⊢ (z = y → (w = ⟨x, z⟩ ↔ w = ⟨x, y⟩))
37		equequ2 1686	. . . . . . . . 9 ⊢ (z = y → (x = z ↔ x = y))
38	36, 37	anbi12d 691	. . . . . . . 8 ⊢ (z = y → ((w = ⟨x, z⟩ ∧ x = z) ↔ (w = ⟨x, y⟩ ∧ x = y)))
39	38	a1i 10	. . . . . . 7 ⊢ (¬ ∀x x = y → (z = y → ((w = ⟨x, z⟩ ∧ x = z) ↔ (w = ⟨x, y⟩ ∧ x = y))))
40	27, 34, 39	cbvexd 2009	. . . . . 6 ⊢ (¬ ∀x x = y → (∃z(w = ⟨x, z⟩ ∧ x = z) ↔ ∃y(w = ⟨x, y⟩ ∧ x = y)))
41	26, 40	exbid 1773	. . . . 5 ⊢ (¬ ∀x x = y → (∃x∃z(w = ⟨x, z⟩ ∧ x = z) ↔ ∃x∃y(w = ⟨x, y⟩ ∧ x = y)))
42	25, 41	pm2.61i 156	. . . 4 ⊢ (∃x∃z(w = ⟨x, z⟩ ∧ x = z) ↔ ∃x∃y(w = ⟨x, y⟩ ∧ x = y))
43	42	abbii 2466	. . 3 ⊢ {w ∣ ∃x∃z(w = ⟨x, z⟩ ∧ x = z)} = {w ∣ ∃x∃y(w = ⟨x, y⟩ ∧ x = y)}
44		df-opab 4624	. . 3 ⊢ {⟨x, z⟩ ∣ x = z} = {w ∣ ∃x∃z(w = ⟨x, z⟩ ∧ x = z)}
45		df-opab 4624	. . 3 ⊢ {⟨x, y⟩ ∣ x = y} = {w ∣ ∃x∃y(w = ⟨x, y⟩ ∧ x = y)}
46	43, 44, 45	3eqtr4i 2383	. 2 ⊢ {⟨x, z⟩ ∣ x = z} = {⟨x, y⟩ ∣ x = y}
47	1, 46	eqtri 2373	1 ⊢ I = {⟨x, y⟩ ∣ x = y}

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  {cab 2339  ⟨cop 4562  {copab 4623   I cid 4764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-opab 4624  df-id 4768

This theorem is referenced by:  dfid2  4770  opabresid  5004