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Theorem breq12 3898
Description: Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
Assertion
Ref Expression
breq12 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝑅𝐶𝐵𝑅𝐷))

Proof of Theorem breq12
StepHypRef Expression
1 breq1 3896 . 2 (𝐴 = 𝐵 → (𝐴𝑅𝐶𝐵𝑅𝐶))
2 breq2 3897 . 2 (𝐶 = 𝐷 → (𝐵𝑅𝐶𝐵𝑅𝐷))
31, 2sylan9bb 455 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝑅𝐶𝐵𝑅𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1312   class class class wbr 3893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-un 3039  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894
This theorem is referenced by:  breq12i  3902  breq12d  3906  breqan12d  3908  posng  4569  isopolem  5675  poxp  6081  rbropapd  6091  ecopover  6479  ecopoverg  6482  ltdcnq  7147  recexpr  7388  ltresr  7568  reapval  8250  ltxr  9449  xrltnr  9453  xrltnsym  9466  xrlttr  9468  xrltso  9469  xrlttri3  9470  xposdif  9552  exmidsbthrlem  12898
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