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Theorem supeq1d 6986
Description: Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypothesis
Ref Expression
supeq1d.1 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
supeq1d (𝜑 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))

Proof of Theorem supeq1d
StepHypRef Expression
1 supeq1d.1 . 2 (𝜑𝐵 = 𝐶)
2 supeq1 6985 . 2 (𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
31, 2syl 14 1 (𝜑 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  supcsup 6981
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 3811  df-sup 6983
This theorem is referenced by:  sup3exmid  8914  supminfex  9597  minmax  11238  xrminmax  11273  xrminrecl  11281  xrminadd  11283  suprzubdc  11953  gcdval  11960  gcdass  12016  pceulem  12294  pceu  12295  pcval  12296  pczpre  12297  pcdiv  12302  pcneg  12324  prdsex  12718  xmetxp  14010  xmetxpbl  14011  txmetcnp  14021  qtopbasss  14024
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