Theorem 3brtr4d 2635

Description: Substitution of equality into both sides of a binary relation.

Hypotheses

Ref	Expression
3brtr4d.1	|- (ph -> ARB)
3brtr4d.2	|- (ph -> C = A)
3brtr4d.3	|- (ph -> D = B)

Assertion

Ref	Expression
3brtr4d	|- (ph -> CRD)

Proof of Theorem 3brtr4d

Step	Hyp	Ref	Expression
1		3brtr4d.1	. 2 |- (ph -> ARB)
2		3brtr4d.2	. . 3 |- (ph -> C = A)
3		3brtr4d.3	. . 3 |- (ph -> D = B)
4	2, 3	breq12d 2621	. 2 |- (ph -> (CRD <-> ARB))
5	1, 4	mpbird 196	1 |- (ph -> CRD)

Syntax hints:   -> wi 3   = wceq 953   class class class wbr 2609

This theorem is referenced by:  lediv12it 5844  recp1lt1 5849  expmwordit 6537  bernneq 6583  absrelet 6804  absimlet 6805  abs2difabst 6840  ser1absdiflem 6866  faclbnd 6882  faclbnd4lem3 6887  faclbnd4lem4 6888  faclbnd6 6891  fsumcmp 6978  fsumabs 6981  climsqueeze 7076  climsqueeze2 7077  ser1cmp 7110  ser1cmp2 7113  cvgcmp2lem 7116  isumcmpi 7150  erelem3 7263  efaddlem14 7293  ef1tllem 7323  ef01tllem2 7326  eflegeolem2 7354  ruclem24 7476  cnmet 7843  dscmet 7856  nvmtri 8238  imsmetlem 8261  nmlnoubi 8388  blometi 8394  ipblnfi 8447  ubthlem11 8470  hhssnv 9054  nmopco 9942  nmopcoadj 9948  idleop 9976  hmopidmch 9990  cdj1 10265  mslb1 10473

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610

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