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Theorem breq123d 5161
Description: Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
Hypotheses
Ref Expression
breq1d.1 (𝜑𝐴 = 𝐵)
breq123d.2 (𝜑𝑅 = 𝑆)
breq123d.3 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
breq123d (𝜑 → (𝐴𝑅𝐶𝐵𝑆𝐷))

Proof of Theorem breq123d
StepHypRef Expression
1 breq1d.1 . . 3 (𝜑𝐴 = 𝐵)
2 breq123d.3 . . 3 (𝜑𝐶 = 𝐷)
31, 2breq12d 5160 . 2 (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐷))
4 breq123d.2 . . 3 (𝜑𝑅 = 𝑆)
54breqd 5158 . 2 (𝜑 → (𝐵𝑅𝐷𝐵𝑆𝐷))
63, 5bitrd 279 1 (𝜑 → (𝐴𝑅𝐶𝐵𝑆𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1536   class class class wbr 5147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148
This theorem is referenced by:  sbcbr123  5201  fmptco  7148  xpsle  17625  invfuc  18030  yonedainv  18337  opphllem3  28771  lmif  28807  islmib  28809  iscgra  28831  isinag  28860  fmptcof2  32673  submomnd  33069  sgnsv  33162  inftmrel  33169  isinftm  33170  submarchi  33175  rlocval  33245  suborng  33324  rprmval  33523  weiunlem1  36444  uncov  37587  iscvlat  39304  paddfval  39779  lhpset  39977  tendofset  40740  diaffval  41012  fnwe2val  43037  aomclem8  43049  afv2eq12d  47164
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