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Theorem 3brtr4d 3835
Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
Hypotheses
Ref Expression
3brtr4d.1  |-  ( ph  ->  A R B )
3brtr4d.2  |-  ( ph  ->  C  =  A )
3brtr4d.3  |-  ( ph  ->  D  =  B )
Assertion
Ref Expression
3brtr4d  |-  ( ph  ->  C R D )

Proof of Theorem 3brtr4d
StepHypRef Expression
1 3brtr4d.1 . 2  |-  ( ph  ->  A R B )
2 3brtr4d.2 . . 3  |-  ( ph  ->  C  =  A )
3 3brtr4d.3 . . 3  |-  ( ph  ->  D  =  B )
42, 3breq12d 3818 . 2  |-  ( ph  ->  ( C R D  <-> 
A R B ) )
51, 4mpbird 165 1  |-  ( ph  ->  C R D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   class class class wbr 3805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806
This theorem is referenced by:  f1oiso2  5518  prarloclemarch2  6741  caucvgprprlemmu  7017  caucvgsrlembound  7102  mulap0  7881  lediv12a  8109  recp1lt1  8114  fldiv4p1lem1div2  9457  intfracq  9472  modqmulnn  9494  addmodlteq  9550  frecfzennn  9578  monoord2  9622  expgt1  9681  leexp2r  9697  leexp1a  9698  bernneq  9760  faclbnd  9835  faclbnd6  9838  facubnd  9839  hashunlem  9898  sqrtgt0  10139  absrele  10188  absimle  10189  abstri  10209  abs2difabs  10213  climsqz  10392  climsqz2  10393  phibnd  10818
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