Theorem dfid3 5025

Description: A stronger version of df-id 5024 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 5024	. 2 |- _I = { <. x , z >. | x = z }
2		ancom 466	. . . . . . . . . . 11 |- ( ( w = <. x , z >. /\ x = z ) <-> ( x = z /\ w = <. x , z >. ) )
3		equcom 1945	. . . . . . . . . . . 12 |- ( x = z <-> z = x )
4	3	anbi1i 731	. . . . . . . . . . 11 |- ( ( x = z /\ w = <. x , z >. ) <-> ( z = x /\ w = <. x , z >. ) )
5	2, 4	bitri 264	. . . . . . . . . 10 |- ( ( w = <. x , z >. /\ x = z ) <-> ( z = x /\ w = <. x , z >. ) )
6	5	exbii 1774	. . . . . . . . 9 |- ( E. z ( w = <. x , z >. /\ x = z ) <-> E. z ( z = x /\ w = <. x , z >. ) )
7		opeq2 4403	. . . . . . . . . . 11 |- ( z = x -> <. x , z >. = <. x , x >. )
8	7	eqeq2d 2632	. . . . . . . . . 10 |- ( z = x -> ( w = <. x , z >. <-> w = <. x , x >. ) )
9	8	equsexvw 1932	. . . . . . . . 9 |- ( E. z ( z = x /\ w = <. x , z >. ) <-> w = <. x , x >. )
10		equid 1939	. . . . . . . . . 10 |- x = x
11	10	biantru 526	. . . . . . . . 9 |- ( w = <. x , x >. <-> ( w = <. x , x >. /\ x = x ) )
12	6, 9, 11	3bitri 286	. . . . . . . 8 |- ( E. z ( w = <. x , z >. /\ x = z ) <-> ( w = <. x , x >. /\ x = x ) )
13	12	exbii 1774	. . . . . . 7 |- ( E. x E. z ( w = <. x , z >. /\ x = z ) <-> E. x ( w = <. x , x >. /\ x = x ) )
14		nfe1 2027	. . . . . . . 8 |- F/ x E. x ( w = <. x , x >. /\ x = x )
15	14	19.9 2072	. . . . . . 7 |- ( E. x E. x ( w = <. x , x >. /\ x = x ) <-> E. x ( w = <. x , x >. /\ x = x ) )
16	13, 15	bitr4i 267	. . . . . 6 |- ( E. x E. z ( w = <. x , z >. /\ x = z ) <-> E. x E. x ( w = <. x , x >. /\ x = x ) )
17		opeq2 4403	. . . . . . . . . . 11 |- ( x = y -> <. x , x >. = <. x , y >. )
18	17	eqeq2d 2632	. . . . . . . . . 10 |- ( x = y -> ( w = <. x , x >. <-> w = <. x , y >. ) )
19		equequ2 1953	. . . . . . . . . 10 |- ( x = y -> ( x = x <-> x = y ) )
20	18, 19	anbi12d 747	. . . . . . . . 9 |- ( x = y -> ( ( w = <. x , x >. /\ x = x ) <-> ( w = <. x , y >. /\ x = y ) ) )
21	20	sps 2055	. . . . . . . 8 |- ( A. x x = y -> ( ( w = <. x , x >. /\ x = x ) <-> ( w = <. x , y >. /\ x = y ) ) )
22	21	drex1 2327	. . . . . . 7 |- ( A. x x = y -> ( E. x ( w = <. x , x >. /\ x = x ) <-> E. y ( w = <. x , y >. /\ x = y ) ) )
23	22	drex2 2328	. . . . . 6 |- ( A. x x = y -> ( E. x E. x ( w = <. x , x >. /\ x = x ) <-> E. x E. y ( w = <. x , y >. /\ x = y ) ) )
24	16, 23	syl5bb 272	. . . . 5 |- ( A. x x = y -> ( E. x E. z ( w = <. x , z >. /\ x = z ) <-> E. x E. y ( w = <. x , y >. /\ x = y ) ) )
25		nfnae 2318	. . . . . 6 |- F/ x -. A. x x = y
26		nfnae 2318	. . . . . . 7 |- F/ y -. A. x x = y
27		nfcvd 2765	. . . . . . . . 9 |- ( -. A. x x = y -> F/_ y w )
28		nfcvf2 2789	. . . . . . . . . 10 |- ( -. A. x x = y -> F/_ y x )
29		nfcvd 2765	. . . . . . . . . 10 |- ( -. A. x x = y -> F/_ y z )
30	28, 29	nfopd 4419	. . . . . . . . 9 |- ( -. A. x x = y -> F/_ y <. x , z >. )
31	27, 30	nfeqd 2772	. . . . . . . 8 |- ( -. A. x x = y -> F/ y w = <. x , z >. )
32	28, 29	nfeqd 2772	. . . . . . . 8 |- ( -. A. x x = y -> F/ y x = z )
33	31, 32	nfand 1826	. . . . . . 7 |- ( -. A. x x = y -> F/ y ( w = <. x , z >. /\ x = z ) )
34		opeq2 4403	. . . . . . . . . 10 |- ( z = y -> <. x , z >. = <. x , y >. )
35	34	eqeq2d 2632	. . . . . . . . 9 |- ( z = y -> ( w = <. x , z >. <-> w = <. x , y >. ) )
36		equequ2 1953	. . . . . . . . 9 |- ( z = y -> ( x = z <-> x = y ) )
37	35, 36	anbi12d 747	. . . . . . . 8 |- ( z = y -> ( ( w = <. x , z >. /\ x = z ) <-> ( w = <. x , y >. /\ x = y ) ) )
38	37	a1i 11	. . . . . . 7 |- ( -. A. x x = y -> ( z = y -> ( ( w = <. x , z >. /\ x = z ) <-> ( w = <. x , y >. /\ x = y ) ) ) )
39	26, 33, 38	cbvexd 2278	. . . . . 6 |- ( -. A. x x = y -> ( E. z ( w = <. x , z >. /\ x = z ) <-> E. y ( w = <. x , y >. /\ x = y ) ) )
40	25, 39	exbid 2091	. . . . 5 |- ( -. A. x x = y -> ( E. x E. z ( w = <. x , z >. /\ x = z ) <-> E. x E. y ( w = <. x , y >. /\ x = y ) ) )
41	24, 40	pm2.61i 176	. . . 4 |- ( E. x E. z ( w = <. x , z >. /\ x = z ) <-> E. x E. y ( w = <. x , y >. /\ x = y ) )
42	41	abbii 2739	. . 3 |- { w | E. x E. z ( w = <. x , z >. /\ x = z ) } = { w | E. x E. y ( w = <. x , y >. /\ x = y ) }
43		df-opab 4713	. . 3 |- { <. x , z >. | x = z } = { w | E. x E. z ( w = <. x , z >. /\ x = z ) }
44		df-opab 4713	. . 3 |- { <. x , y >. | x = y } = { w | E. x E. y ( w = <. x , y >. /\ x = y ) }
45	42, 43, 44	3eqtr4i 2654	. 2 |- { <. x , z >. | x = z } = { <. x , y >. | x = y }
46	1, 45	eqtri 2644	1 |- _I = { <. x , y >. | x = y }

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   {cab 2608   <.cop 4183   {copab 4712   _I cid 5023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-id 5024

This theorem is referenced by:  dfid2  5027  reli  5249  opabresid  5455  ider  7779  cnmptid  21464