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 |- (E.z(w = <.x, z>. /\ x = z) <-> E.z(z = x /\ w = <.x, z>.))
7		vex 2863	. . . . . . . . . 10 |- x e. _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 |- (E.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 |- (E.z(w = <.x, z>. /\ x = z) <-> (w = <.x, x>. /\ x = x))
14	13	exbii 1582	. . . . . . 7 |- (E.xE.z(w = <.x, z>. /\ x = z) <-> E.x(w = <.x, x>. /\ x = x))
15		nfe1 1732	. . . . . . . 8 |- F/xE.x(w = <.x, x>. /\ x = x)
16	15	19.9 1783	. . . . . . 7 |- (E.xE.x(w = <.x, x>. /\ x = x) <-> E.x(w = <.x, x>. /\ x = x))
17	14, 16	bitr4i 243	. . . . . 6 |- (E.xE.z(w = <.x, z>. /\ x = z) <-> E.xE.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 |- (A.x x = y -> ((w = <.x, x>. /\ x = x) <-> (w = <.x, y>. /\ x = y)))
23	22	drex1 1967	. . . . . . 7 |- (A.x x = y -> (E.x(w = <.x, x>. /\ x = x) <-> E.y(w = <.x, y>. /\ x = y)))
24	23	drex2 1968	. . . . . 6 |- (A.x x = y -> (E.xE.x(w = <.x, x>. /\ x = x) <-> E.xE.y(w = <.x, y>. /\ x = y)))
25	17, 24	syl5bb 248	. . . . 5 |- (A.x x = y -> (E.xE.z(w = <.x, z>. /\ x = z) <-> E.xE.y(w = <.x, y>. /\ x = y)))
26		nfnae 1956	. . . . . 6 |- F/x -. A.x x = y
27		nfnae 1956	. . . . . . 7 |- F/y -. A.x x = y
28		nfcvd 2491	. . . . . . . . 9 |- (-. A.x x = y -> F/_yw)
29		nfcvf2 2513	. . . . . . . . . 10 |- (-. A.x x = y -> F/_yx)
30		nfcvd 2491	. . . . . . . . . 10 |- (-. A.x x = y -> F/_yz)
31	29, 30	nfopd 4606	. . . . . . . . 9 |- (-. A.x x = y -> F/_y<.x, z>.)
32	28, 31	nfeqd 2504	. . . . . . . 8 |- (-. A.x x = y -> F/y w = <.x, z>.)
33	29, 30	nfeqd 2504	. . . . . . . 8 |- (-. A.x x = y -> F/y x = z)
34	32, 33	nfand 1822	. . . . . . 7 |- (-. A.x x = y -> F/y(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 |- (-. A.x x = y -> (z = y -> ((w = <.x, z>. /\ x = z) <-> (w = <.x, y>. /\ x = y))))
40	27, 34, 39	cbvexd 2009	. . . . . 6 |- (-. A.x x = y -> (E.z(w = <.x, z>. /\ x = z) <-> E.y(w = <.x, y>. /\ x = y)))
41	26, 40	exbid 1773	. . . . 5 |- (-. A.x x = y -> (E.xE.z(w = <.x, z>. /\ x = z) <-> E.xE.y(w = <.x, y>. /\ x = y)))
42	25, 41	pm2.61i 156	. . . 4 |- (E.xE.z(w = <.x, z>. /\ x = z) <-> E.xE.y(w = <.x, y>. /\ x = y))
43	42	abbii 2466	. . 3 |- {w | E.xE.z(w = <.x, z>. /\ x = z)} = {w | E.xE.y(w = <.x, y>. /\ x = y)}
44		df-opab 4624	. . 3 |- {<.x, z>. | x = z} = {w | E.xE.z(w = <.x, z>. /\ x = z)}
45		df-opab 4624	. . 3 |- {<.x, y>. | x = y} = {w | E.xE.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  A.wal 1540  E.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