MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  releqi Unicode version

Theorem releqi 4772
Description: Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)
Hypothesis
Ref Expression
releqi.1  |-  A  =  B
Assertion
Ref Expression
releqi  |-  ( Rel 
A  <->  Rel  B )

Proof of Theorem releqi
StepHypRef Expression
1 releqi.1 . 2  |-  A  =  B
2 releq 4771 . 2  |-  ( A  =  B  ->  ( Rel  A  <->  Rel  B ) )
31, 2ax-mp 8 1  |-  ( Rel 
A  <->  Rel  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623   Rel wrel 4694
This theorem is referenced by:  reliun  4806  reluni  4808  relint  4809  reldmmpt2  5955  tfrlem6  6398  relsdom  6870  cda1dif  7802  0rest  13334  firest  13337  2oppchomf  13627  xpcbas  13952  oppchofcl  14034  oyoncl  14044  releqg  14664  reldvdsr  15426  eltopspOLD  16656  restbas  16889  hlimcaui  21816  relbigcup  24437  fnsingle  24458  funimage  24467  colinrel  24680
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166  df-rel 4696
  Copyright terms: Public domain W3C validator