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Theorem breq12 3760
Description: Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
Assertion
Ref Expression
breq12 ((A = B 𝐶 = 𝐷) → (A𝑅𝐶B𝑅𝐷))

Proof of Theorem breq12
StepHypRef Expression
1 breq1 3758 . 2 (A = B → (A𝑅𝐶B𝑅𝐶))
2 breq2 3759 . 2 (𝐶 = 𝐷 → (B𝑅𝐶B𝑅𝐷))
31, 2sylan9bb 435 1 ((A = B 𝐶 = 𝐷) → (A𝑅𝐶B𝑅𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   class class class wbr 3755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756
This theorem is referenced by:  breq12i  3764  breq12d  3768  breqan12d  3770  posng  4355  isopolem  5404  poxp  5794  isprmpt2  5799  ecopover  6140  ecopoverg  6143  ltdcnq  6381  recexpr  6609  ltresr  6716  reapval  7340  ltxr  8445  xrltnr  8451  xrltnsym  8464  xrlttr  8466  xrltso  8467  xrlttri3  8468
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