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Theorem supeq1d 6932
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 6931 . 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 1335  supcsup 6927
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-uni 3774  df-sup 6929
This theorem is referenced by:  sup3exmid  8829  supminfex  9509  minmax  11133  xrminmax  11166  xrminrecl  11174  xrminadd  11176  gcdval  11847  gcdass  11903  xmetxp  12949  xmetxpbl  12950  txmetcnp  12960  qtopbasss  12963
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