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Theorem dfid3 4501
 Description: A stronger version of df-id 4500 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 4500 . 2
2 ancom 439 . . . . . . . . . . 11
3 equcom 1693 . . . . . . . . . . . 12
43anbi1i 678 . . . . . . . . . . 11
52, 4bitri 242 . . . . . . . . . 10
65exbii 1593 . . . . . . . . 9
7 vex 2961 . . . . . . . . . 10
8 opeq2 3987 . . . . . . . . . . 11
98eqeq2d 2449 . . . . . . . . . 10
107, 9ceqsexv 2993 . . . . . . . . 9
11 equid 1689 . . . . . . . . . 10
1211biantru 493 . . . . . . . . 9
136, 10, 123bitri 264 . . . . . . . 8
1413exbii 1593 . . . . . . 7
15 nfe1 1748 . . . . . . . 8
161519.9 1798 . . . . . . 7
1714, 16bitr4i 245 . . . . . 6
18 opeq2 3987 . . . . . . . . . . 11
1918eqeq2d 2449 . . . . . . . . . 10
20 equequ2 1699 . . . . . . . . . 10
2119, 20anbi12d 693 . . . . . . . . 9
2221sps 1771 . . . . . . . 8
2322drex1 2060 . . . . . . 7
2423drex2 2061 . . . . . 6
2517, 24syl5bb 250 . . . . 5
26 nfnae 2045 . . . . . 6
27 nfnae 2045 . . . . . . 7
28 nfcvd 2575 . . . . . . . . 9
29 nfcvf2 2597 . . . . . . . . . 10
30 nfcvd 2575 . . . . . . . . . 10
3129, 30nfopd 4003 . . . . . . . . 9
3228, 31nfeqd 2588 . . . . . . . 8
3329, 30nfeqd 2588 . . . . . . . 8
3432, 33nfand 1844 . . . . . . 7
35 opeq2 3987 . . . . . . . . . 10
3635eqeq2d 2449 . . . . . . . . 9
37 equequ2 1699 . . . . . . . . 9
3836, 37anbi12d 693 . . . . . . . 8
3938a1i 11 . . . . . . 7
4027, 34, 39cbvexd 1989 . . . . . 6
4126, 40exbid 1790 . . . . 5
4225, 41pm2.61i 159 . . . 4
4342abbii 2550 . . 3
44 df-opab 4269 . . 3
45 df-opab 4269 . . 3
4643, 44, 453eqtr4i 2468 . 2
471, 46eqtri 2458 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wa 360  wal 1550  wex 1551   wceq 1653  cab 2424  cop 3819  copab 4267   cid 4495 This theorem is referenced by:  dfid2  4502  reli  5004  opabresid  5196  ider  6941  cnmptid  17695 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4269  df-id 4500
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