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Theorem supeq1i 6868
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 6866 . 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 1331  supcsup 6862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-uni 3732  df-sup 6864
This theorem is referenced by:  infrenegsupex  9382  maxcom  10968  xrmax2sup  11016  xrmaxltsup  11020  xrmaxadd  11023  infxrnegsupex  11025  gcdcom  11651  gcdass  11692
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