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Theorem breq123d 5162
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 5161 . 2 (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐷))
4 breq123d.2 . . 3 (𝜑𝑅 = 𝑆)
54breqd 5159 . 2 (𝜑 → (𝐵𝑅𝐷𝐵𝑆𝐷))
63, 5bitrd 278 1 (𝜑 → (𝐴𝑅𝐶𝐵𝑆𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541   class class class wbr 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149
This theorem is referenced by:  sbcbr123  5202  fmptco  7129  xpsle  17529  invfuc  17931  yonedainv  18238  opphllem3  28255  lmif  28291  islmib  28293  iscgra  28315  isinag  28344  fmptcof2  32137  submomnd  32486  sgnsv  32577  inftmrel  32584  isinftm  32585  submarchi  32590  suborng  32691  rprmval  32895  uncov  36772  iscvlat  38496  paddfval  38971  lhpset  39169  tendofset  39932  diaffval  40204  fnwe2val  42093  aomclem8  42105  afv2eq12d  46222
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