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

Theorem releqd 4728
Description: Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)
Hypothesis
Ref Expression
releqd.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
releqd  |-  ( ph  ->  ( Rel  A  <->  Rel  B ) )

Proof of Theorem releqd
StepHypRef Expression
1 releqd.1 . 2  |-  ( ph  ->  A  =  B )
2 releq 4726 . 2  |-  ( A  =  B  ->  ( Rel  A  <->  Rel  B ) )
31, 2syl 14 1  |-  ( ph  ->  ( Rel  A  <->  Rel  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1364   Rel wrel 4649
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-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157  df-rel 4651
This theorem is referenced by:  dftpos3  6288  tposfo2  6293  tposf12  6295  imasaddfnlemg  12794  releqgg  13176  dvdsrd  13461  isunitd  13473  lmreltop  14170  cnprcl2k  14183
  Copyright terms: Public domain W3C validator