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Theorem supeq1d 6589
Description: Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypothesis
Ref Expression
supeq1d.1  |-  ( ph  ->  B  =  C )
Assertion
Ref Expression
supeq1d  |-  ( ph  ->  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R )
)

Proof of Theorem supeq1d
StepHypRef Expression
1 supeq1d.1 . 2  |-  ( ph  ->  B  =  C )
2 supeq1 6588 . 2  |-  ( B  =  C  ->  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R ) )
31, 2syl 14 1  |-  ( ph  ->  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   supcsup 6584
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 3628  df-sup 6586
This theorem is referenced by:  supminfex  8980  minmax  10486  gcdval  10731  gcdass  10784
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