Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfid3 Unicode version

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.
StepHypRef Expression
1 df-id 4800 . 2
2 ancom 450 . . . . . . . . . . 11
3 equcom 1794 . . . . . . . . . . . 12
43anbi1i 695 . . . . . . . . . . 11
52, 4bitri 249 . . . . . . . . . 10
65exbii 1667 . . . . . . . . 9
7 vex 3112 . . . . . . . . . 10
8 opeq2 4218 . . . . . . . . . . 11
98eqeq2d 2471 . . . . . . . . . 10
107, 9ceqsexv 3146 . . . . . . . . 9
11 equid 1791 . . . . . . . . . 10
1211biantru 505 . . . . . . . . 9
136, 10, 123bitri 271 . . . . . . . 8
1413exbii 1667 . . . . . . 7
15 nfe1 1840 . . . . . . . 8
161519.9 1893 . . . . . . 7
1714, 16bitr4i 252 . . . . . 6
18 opeq2 4218 . . . . . . . . . . 11
1918eqeq2d 2471 . . . . . . . . . 10
20 equequ2 1799 . . . . . . . . . 10
2119, 20anbi12d 710 . . . . . . . . 9
2221sps 1865 . . . . . . . 8
2322drex1 2069 . . . . . . 7
2423drex2 2070 . . . . . 6
2517, 24syl5bb 257 . . . . 5
26 nfnae 2058 . . . . . 6
27 nfnae 2058 . . . . . . 7
28 nfcvd 2620 . . . . . . . . 9
29 nfcvf2 2645 . . . . . . . . . 10
30 nfcvd 2620 . . . . . . . . . 10
3129, 30nfopd 4234 . . . . . . . . 9
3228, 31nfeqd 2626 . . . . . . . 8
3329, 30nfeqd 2626 . . . . . . . 8
3432, 33nfand 1925 . . . . . . 7
35 opeq2 4218 . . . . . . . . . 10
3635eqeq2d 2471 . . . . . . . . 9
37 equequ2 1799 . . . . . . . . 9
3836, 37anbi12d 710 . . . . . . . 8
3938a1i 11 . . . . . . 7
4027, 34, 39cbvexd 2026 . . . . . 6
4126, 40exbid 1886 . . . . 5
4225, 41pm2.61i 164 . . . 4
4342abbii 2591 . . 3
44 df-opab 4511 . . 3
45 df-opab 4511 . . 3
4643, 44, 453eqtr4i 2496 . 2
471, 46eqtri 2486 1
 Colors of variables: wff setvar class 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
 Copyright terms: Public domain W3C validator