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Theorem breq123d 4699
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 4698 . 2 (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐷))
4 breq123d.2 . . 3 (𝜑𝑅 = 𝑆)
54breqd 4696 . 2 (𝜑 → (𝐵𝑅𝐷𝐵𝑆𝐷))
63, 5bitrd 268 1 (𝜑 → (𝐴𝑅𝐶𝐵𝑆𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523   class class class wbr 4685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686
This theorem is referenced by:  sbcbr123  4739  fmptco  6436  xpsle  16288  invfuc  16681  yonedainv  16968  opphllem3  25686  lmif  25722  islmib  25724  iscgra  25746  isinag  25774  fmptcof2  29585  submomnd  29838  sgnsv  29855  inftmrel  29862  isinftm  29863  submarchi  29868  suborng  29943  uncov  33520  iscvlat  34928  paddfval  35401  lhpset  35599  tendofset  36363  diaffval  36636  fnwe2val  37936  aomclem8  37948
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