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Theorem distel 25433
 Description: Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 4383 and elirrv 7567.) (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 4383 . . 3
2 df-ex 1552 . . . 4
3 nfnae 2045 . . . . . 6
4 dveel1 2107 . . . . . . . 8
53, 4nfd 1783 . . . . . . 7
65nfnd 1810 . . . . . 6
7 elequ2 1731 . . . . . . . 8
87notbid 287 . . . . . . 7
98a1i 11 . . . . . 6
103, 6, 9cbvald 1987 . . . . 5
1110notbid 287 . . . 4
122, 11syl5bb 250 . . 3
131, 12mpbii 204 . 2
14 elirrv 7567 . . . . 5
15 elequ1 1729 . . . . 5
1614, 15mtbii 295 . . . 4
1716alimi 1569 . . 3
1817con3i 130 . 2
1913, 18impbii 182 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178  wal 1550  wex 1551 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-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-reg 7562 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  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-ne 2603  df-ral 2712  df-rex 2713  df-v 2960  df-dif 3325  df-un 3327  df-nul 3631  df-sn 3822  df-pr 3823
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