ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  supeq1d Unicode version

Theorem supeq1d 7089
Description: Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypothesis
Ref Expression
supeq1d.1  |-  ( ph  ->  B  =  C )
Assertion
Ref Expression
supeq1d  |-  ( ph  ->  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R )
)

Proof of Theorem supeq1d
StepHypRef Expression
1 supeq1d.1 . 2  |-  ( ph  ->  B  =  C )
2 supeq1 7088 . 2  |-  ( B  =  C  ->  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R ) )
31, 2syl 14 1  |-  ( ph  ->  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1373   supcsup 7084
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187
This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-uni 3851  df-sup 7086
This theorem is referenced by:  sup3exmid  9030  supminfex  9718  suprzubdc  10379  minmax  11541  xrminmax  11576  xrminrecl  11584  xrminadd  11586  gcdval  12280  gcdass  12336  pceulem  12617  pceu  12618  pcval  12619  pczpre  12620  pcdiv  12625  pcneg  12648  prdsex  13101  prdsval  13105  xmetxp  14979  xmetxpbl  14980  txmetcnp  14990  qtopbasss  14993  hovera  15119  hoverb  15120  hoverlt1  15121  hovergt0  15122  ivthdich  15125
  Copyright terms: Public domain W3C validator