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Theorem supeq1i 6986
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 6984 . 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 set class
Syntax hints:    = wceq 1353   supcsup 6980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-uni 3810  df-sup 6982
This theorem is referenced by:  infrenegsupex  9592  maxcom  11207  xrmax2sup  11257  xrmaxltsup  11261  xrmaxadd  11264  infxrnegsupex  11266  gcdcom  11968  gcdass  12010
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