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Theorem supeq1 7450
 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 2904 . . . . 5
2 rexeq 2905 . . . . . . 7
32imbi2d 308 . . . . . 6
43ralbidv 2725 . . . . 5
51, 4anbi12d 692 . . . 4
65rabbidv 2948 . . 3
76unieqd 4026 . 2
8 df-sup 7446 . 2
9 df-sup 7446 . 2
107, 8, 93eqtr4g 2493 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359   wceq 1652  wral 2705  wrex 2706  crab 2709  cuni 4015   class class class wbr 4212  csup 7445 This theorem is referenced by:  supeq1d  7451  supeq1i  7452  ramcl2lem  13377  odval  15172  submod  15203  bndth  18983  ioorval  19466  uniioombllem6  19480  mdegcl  19992 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-uni 4016  df-sup 7446
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