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Theorem dfid3 4768
 Description: A stronger version of df-id 4767 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 4767 . 2
2 ancom 437 . . . . . . . . . . 11
3 equcom 1680 . . . . . . . . . . . 12
43anbi1i 676 . . . . . . . . . . 11
52, 4bitri 240 . . . . . . . . . 10
65exbii 1582 . . . . . . . . 9
7 vex 2862 . . . . . . . . . 10
8 opeq2 4579 . . . . . . . . . . 11
98eqeq2d 2364 . . . . . . . . . 10
107, 9ceqsexv 2894 . . . . . . . . 9
11 equid 1676 . . . . . . . . . 10
1211biantru 491 . . . . . . . . 9
136, 10, 123bitri 262 . . . . . . . 8
1413exbii 1582 . . . . . . 7
15 nfe1 1732 . . . . . . . 8
161519.9 1783 . . . . . . 7
1714, 16bitr4i 243 . . . . . 6
18 opeq2 4579 . . . . . . . . . . 11
1918eqeq2d 2364 . . . . . . . . . 10
20 equequ2 1686 . . . . . . . . . 10
2119, 20anbi12d 691 . . . . . . . . 9
2221sps 1754 . . . . . . . 8
2322drex1 1967 . . . . . . 7
2423drex2 1968 . . . . . 6
2517, 24syl5bb 248 . . . . 5
26 nfnae 1956 . . . . . 6
27 nfnae 1956 . . . . . . 7
28 nfcvd 2490 . . . . . . . . 9
29 nfcvf2 2512 . . . . . . . . . 10
30 nfcvd 2490 . . . . . . . . . 10
3129, 30nfopd 4605 . . . . . . . . 9
3228, 31nfeqd 2503 . . . . . . . 8
3329, 30nfeqd 2503 . . . . . . . 8
3432, 33nfand 1822 . . . . . . 7
35 opeq2 4579 . . . . . . . . . 10
3635eqeq2d 2364 . . . . . . . . 9
37 equequ2 1686 . . . . . . . . 9
3836, 37anbi12d 691 . . . . . . . 8
3938a1i 10 . . . . . . 7
4027, 34, 39cbvexd 2009 . . . . . 6
4126, 40exbid 1773 . . . . 5
4225, 41pm2.61i 156 . . . 4
4342abbii 2465 . . 3
44 df-opab 4623 . . 3
45 df-opab 4623 . . 3
4643, 44, 453eqtr4i 2383 . 2
471, 46eqtri 2373 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 176   wa 358  wal 1540  wex 1541   wceq 1642  cab 2339  cop 4561  copab 4622   cid 4763 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-id 4767 This theorem is referenced by:  dfid2  4769  opabresid  5003
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