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

Theorem supeq1d 7088
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 7087 . 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 1372   supcsup 7083
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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186
This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-uni 3850  df-sup 7085
This theorem is referenced by:  sup3exmid  9029  supminfex  9717  suprzubdc  10377  minmax  11512  xrminmax  11547  xrminrecl  11555  xrminadd  11557  gcdval  12251  gcdass  12307  pceulem  12588  pceu  12589  pcval  12590  pczpre  12591  pcdiv  12596  pcneg  12619  prdsex  13072  prdsval  13076  xmetxp  14950  xmetxpbl  14951  txmetcnp  14961  qtopbasss  14964  hovera  15090  hoverb  15091  hoverlt1  15092  hovergt0  15093  ivthdich  15096
  Copyright terms: Public domain W3C validator