MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  breq12i Unicode version

Theorem breq12i 4181
Description: Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
Hypotheses
Ref Expression
breq1i.1  |-  A  =  B
breq12i.2  |-  C  =  D
Assertion
Ref Expression
breq12i  |-  ( A R C  <->  B R D )

Proof of Theorem breq12i
StepHypRef Expression
1 breq1i.1 . 2  |-  A  =  B
2 breq12i.2 . 2  |-  C  =  D
3 breq12 4177 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A R C  <-> 
B R D ) )
41, 2, 3mp2an 654 1  |-  ( A R C  <->  B R D )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649   class class class wbr 4172
This theorem is referenced by:  3brtr3g  4203  3brtr4g  4204  caovord2  6218  domunfican  7338  ltsonq  8802  ltanq  8804  ltmnq  8805  prlem934  8866  prlem936  8880  ltsosr  8925  ltasr  8931  ltneg  9484  leneg  9487  inelr  9946  lt2sqi  11425  le2sqi  11426  nn0le2msqi  11515  mdsldmd1i  23787  divcnvlin  25165  axlowdimlem6  25790
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173
  Copyright terms: Public domain W3C validator