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Theorem breq12 3732
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 3730 . 2 (A = B → (A𝑅𝐶B𝑅𝐶))
2 breq2 3731 . 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 1223   class class class wbr 3727
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-v 2528  df-un 2890  df-sn 3345  df-pr 3346  df-op 3348  df-br 3728
This theorem is referenced by:  breq12i  3736  breq12d  3740  breqan12d  3742  posng  4327  isopolem  5374  poxp  5763  isprmpt2  5768  ecopover  6103  ecopoverg  6106  ltdcnq  6242  recexpr  6459  ltresr  6545  reapval  7163
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