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Theorem supeq1i 7800
Description: Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypothesis
Ref Expression
supeq1i.1  |-  B  =  C
Assertion
Ref Expression
supeq1i  |-  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R )

Proof of Theorem supeq1i
StepHypRef Expression
1 supeq1i.1 . 2  |-  B  =  C
2 supeq1 7798 . 2  |-  ( B  =  C  ->  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R ) )
31, 2ax-mp 5 1  |-  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   supcsup 7793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-uni 4192  df-sup 7794
This theorem is referenced by:  supsn  7822  infmsup  10411  nninfm  11038  nn0infm  11039  supxrmnf  11383  rpsup  11808  resup  11809  gcdcom  13808  gcdass  13833  imasdsval2  14558  imasdsf1olem  20066  ovolgelb  21081  itg2seq  21338  itg2i1fseq  21351  itg2cnlem1  21357  dvfsumrlim  21621  pserdvlem2  22011  logtayl  22223  ftalem6  22533  nmopnegi  25506  nmop0  25527  nmfn0  25528  esumnul  26638  ismblfin  28572  ovoliunnfl  28573  voliunnfl  28575  itg2addnclem  28583
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