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Theorem supeq1d 6964
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 6963 . 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 1348  supcsup 6959
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-uni 3797  df-sup 6961
This theorem is referenced by:  sup3exmid  8873  supminfex  9556  minmax  11193  xrminmax  11228  xrminrecl  11236  xrminadd  11238  suprzubdc  11907  gcdval  11914  gcdass  11970  pceulem  12248  pceu  12249  pcval  12250  pczpre  12251  pcdiv  12256  pcneg  12278  xmetxp  13301  xmetxpbl  13302  txmetcnp  13312  qtopbasss  13315
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