Theorem breqtrdi 4124

Description: A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)

Hypotheses

Ref	Expression
breqtrdi.1	|- ( ph -> A R B )
breqtrdi.2	|- B = C

Assertion

Ref	Expression
breqtrdi	|- ( ph -> A R C )

Proof of Theorem breqtrdi

Step	Hyp	Ref	Expression
1		breqtrdi.1	. 2 |- ( ph -> A R B )
2		eqid 2229	. 2 |- A = A
3		breqtrdi.2	. 2 |- B = C
4	1, 2, 3	3brtr3g 4116	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1395   class class class wbr 4083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084

This theorem is referenced by:  breqtrrdi  4125  en2eleq  7384  en2other2  7385  dju0en  7407  ltm1sr  7975  maxle2  11739  xrmax2sup  11781  mertenslem2  12063  ege2le3  12198  cos01gt0  12290  sin02gt0  12291  cos12dec  12295  bitsfzolem  12481  bitsmod  12483  unennn  12984  dvef  15417  sin0pilem2  15472  cosq23lt0  15523  cosq34lt1  15540  cos02pilt1  15541  logbgcd1irraplemexp  15658  lgslem3  15697  lgsquadlem1  15772  lgsquadlem3  15774  trilpolemeq1  16496