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Theorem breq123d 4001
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 4000 . 2 (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐷))
4 breq123d.2 . . 3 (𝜑𝑅 = 𝑆)
54breqd 3998 . 2 (𝜑 → (𝐵𝑅𝐷𝐵𝑆𝐷))
63, 5bitrd 187 1 (𝜑 → (𝐴𝑅𝐶𝐵𝑆𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1348   class class class wbr 3987
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988
This theorem is referenced by:  sbcbrg  4041  fmptco  5660
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