Theorem dfid3 4801

Description: A stronger version of df-id 4800 that doesn't require and 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

Proof of Theorem dfid3

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-id 4800	. 2
2		ancom 450	. . . . . . . . . . 11
3		equcom 1794	. . . . . . . . . . . 12
4	3	anbi1i 695	. . . . . . . . . . 11
5	2, 4	bitri 249	. . . . . . . . . 10
6	5	exbii 1667	. . . . . . . . 9
7		vex 3112	. . . . . . . . . 10
8		opeq2 4218	. . . . . . . . . . 11
9	8	eqeq2d 2471	. . . . . . . . . 10
10	7, 9	ceqsexv 3146	. . . . . . . . 9
11		equid 1791	. . . . . . . . . 10
12	11	biantru 505	. . . . . . . . 9
13	6, 10, 12	3bitri 271	. . . . . . . 8
14	13	exbii 1667	. . . . . . 7
15		nfe1 1840	. . . . . . . 8
16	15	19.9 1893	. . . . . . 7
17	14, 16	bitr4i 252	. . . . . 6
18		opeq2 4218	. . . . . . . . . . 11
19	18	eqeq2d 2471	. . . . . . . . . 10
20		equequ2 1799	. . . . . . . . . 10
21	19, 20	anbi12d 710	. . . . . . . . 9
22	21	sps 1865	. . . . . . . 8
23	22	drex1 2069	. . . . . . 7
24	23	drex2 2070	. . . . . 6
25	17, 24	syl5bb 257	. . . . 5
26		nfnae 2058	. . . . . 6
27		nfnae 2058	. . . . . . 7
28		nfcvd 2620	. . . . . . . . 9
29		nfcvf2 2645	. . . . . . . . . 10
30		nfcvd 2620	. . . . . . . . . 10
31	29, 30	nfopd 4234	. . . . . . . . 9
32	28, 31	nfeqd 2626	. . . . . . . 8
33	29, 30	nfeqd 2626	. . . . . . . 8
34	32, 33	nfand 1925	. . . . . . 7
35		opeq2 4218	. . . . . . . . . 10
36	35	eqeq2d 2471	. . . . . . . . 9
37		equequ2 1799	. . . . . . . . 9
38	36, 37	anbi12d 710	. . . . . . . 8
39	38	a1i 11	. . . . . . 7
40	27, 34, 39	cbvexd 2026	. . . . . 6
41	26, 40	exbid 1886	. . . . 5
42	25, 41	pm2.61i 164	. . . 4
43	42	abbii 2591	. . 3
44		df-opab 4511	. . 3
45		df-opab 4511	. . 3
46	43, 44, 45	3eqtr4i 2496	. 2
47	1, 46	eqtri 2486	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  {cab 2442  <.cop 4035  {copab 4509  cid 4795

This theorem is referenced by:  dfid2  4802  reli  5135  opabresid  5332  ider  7364  cnmptid  20162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-opab 4511  df-id 4800