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Theorem supeq1 6493
 Description: Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
Assertion
Ref Expression
supeq1

Proof of Theorem supeq1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 raleq 2554 . . . . 5
2 rexeq 2555 . . . . . . 7
32imbi2d 228 . . . . . 6
43ralbidv 2373 . . . . 5
51, 4anbi12d 457 . . . 4
65rabbidv 2599 . . 3
76unieqd 3632 . 2
8 df-sup 6491 . 2
9 df-sup 6491 . 2
107, 8, 93eqtr4g 2140 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 102   wceq 1285  wral 2353  wrex 2354  crab 2357  cuni 3621   class class class wbr 3805  csup 6489 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-uni 3622  df-sup 6491 This theorem is referenced by:  supeq1d  6494  supeq1i  6495  infeq1  6518
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