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Theorem supeq1i 6460
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 6458 . 2 (𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
31, 2ax-mp 7 1 sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)
Colors of variables: wff set class
Syntax hints:   = wceq 1285  supcsup 6454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-uni 3610  df-sup 6456
This theorem is referenced by:  infrenegsupex  8763  maxcom  10227  gcdcom  10509  gcdass  10548
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