Theorem breq12 3798

Description: Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)

Assertion

Ref	Expression
breq12	|- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Proof of Theorem breq12

Step	Hyp	Ref	Expression
1		breq1 3796	. 2 |- ( A = B -> ( A R C <-> B R C ) )
2		breq2 3797	. 2 |- ( C = D -> ( B R C <-> B R D ) )
3	1, 2	sylan9bb 450	1 |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   class class class wbr 3793

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794

This theorem is referenced by:  breq12i  3802  breq12d  3806  breqan12d  3808  posng  4438  isopolem  5492  poxp  5884  isprmpt2  5892  ecopover  6270  ecopoverg  6273  ltdcnq  6649  recexpr  6890  ltresr  7069  reapval  7743  ltxr  8927  xrltnr  8931  xrltnsym  8944  xrlttr  8946  xrltso  8947  xrlttri3  8948