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Theorem breq123d 3913
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 3912 . 2 (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐷))
4 breq123d.2 . . 3 (𝜑𝑅 = 𝑆)
54breqd 3910 . 2 (𝜑 → (𝐵𝑅𝐷𝐵𝑆𝐷))
63, 5bitrd 187 1 (𝜑 → (𝐴𝑅𝐶𝐵𝑆𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1316   class class class wbr 3899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900
This theorem is referenced by:  sbcbrg  3952  fmptco  5554
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