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Theorem distel 24160
 Description: Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 4192 and elirrv 7311.) (Contributed by Scott Fenton, 15-Dec-2010.)
Assertion
Ref Expression
distel

Proof of Theorem distel
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 el 4192 . . 3
2 df-ex 1529 . . . 4
3 nfnae 1896 . . . . . 6
4 dveel1 1959 . . . . . . . 8
53, 4nfd 1746 . . . . . . 7
65nfnd 1760 . . . . . 6
7 elequ2 1689 . . . . . . . 8
87notbid 285 . . . . . . 7
98a1i 10 . . . . . 6
103, 6, 9cbvald 1948 . . . . 5
1110notbid 285 . . . 4
122, 11syl5bb 248 . . 3
131, 12mpbii 202 . 2
14 elirrv 7311 . . . . 5
15 elequ1 1687 . . . . 5
1614, 15mtbii 293 . . . 4
1716alimi 1546 . . 3
1817con3i 127 . 2
1913, 18impbii 180 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 176  wal 1527  wex 1528   wceq 1623   wcel 1684 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-reg 7306 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-nul 3456  df-sn 3646  df-pr 3647
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