MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  breqan12d Unicode version

Theorem breqan12d 4038
Description: Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
Hypotheses
Ref Expression
breq1d.1  |-  ( ph  ->  A  =  B )
breqan12i.2  |-  ( ps 
->  C  =  D
)
Assertion
Ref Expression
breqan12d  |-  ( (
ph  /\  ps )  ->  ( A R C  <-> 
B R D ) )

Proof of Theorem breqan12d
StepHypRef Expression
1 breq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 breqan12i.2 . 2  |-  ( ps 
->  C  =  D
)
3 breq12 4028 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A R C  <-> 
B R D ) )
41, 2, 3syl2an 463 1  |-  ( (
ph  /\  ps )  ->  ( A R C  <-> 
B R D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623   class class class wbr 4023
This theorem is referenced by:  breqan12rd  4039  soisores  5824  isoid  5826  isores3  5832  isoini2  5836  ofrfval  6086  fnwelem  6230  fnse  6232  ovec  6768  wemaplem1  7261  r0weon  7640  sornom  7903  enqbreq2  8544  nqereu  8553  ordpinq  8567  lterpq  8594  ltresr2  8763  axpre-ltadd  8789  leltadd  9258  lemul1a  9610  negiso  9730  xltneg  10544  lt2sq  11177  le2sq  11178  sqrle  11746  prdsleval  13376  efgcpbllema  15063  icopnfhmeo  18441  iccpnfhmeo  18443  xrhmeo  18444  reefiso  19824  sinord  19896  logltb  19953  logccv  20010  atanord  20223  birthdaylem3  20248  lgsquadlem3  20595  mddmd  22881  erdszelem4  23725  erdszelem8  23729  cgrextend  24631  monotuz  27026  monotoddzzfi  27027  expmordi  27032  wepwsolem  27138  fnwe2val  27146  aomclem8  27159  idlaut  30285
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-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024
  Copyright terms: Public domain W3C validator