Theorem breq12 3769

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 3767	. 2 |- ( A = B -> ( A R C <-> B R C ) )
2		breq2 3768	. 2 |- ( C = D -> ( B R C <-> B R D ) )
3	1, 2	sylan9bb 435	1 |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   class class class wbr 3764

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765

This theorem is referenced by:  breq12i  3773  breq12d  3777  breqan12d  3779  posng  4412  isopolem  5461  poxp  5853  isprmpt2  5858  ecopover  6204  ecopoverg  6207  ltdcnq  6495  recexpr  6736  ltresr  6915  reapval  7567  ltxr  8695  xrltnr  8701  xrltnsym  8714  xrlttr  8716  xrltso  8717  xrlttri3  8718