Theorem breqtrdi 4022

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 2165	. 2 |- A = A
3		breqtrdi.2	. 2 |- B = C
4	1, 2, 3	3brtr3g 4014	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1343   class class class wbr 3981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727  df-un 3119  df-sn 3581  df-pr 3582  df-op 3584  df-br 3982

This theorem is referenced by:  breqtrrdi  4023  en2eleq  7147  en2other2  7148  dju0en  7166  ltm1sr  7714  maxle2  11150  xrmax2sup  11191  mertenslem2  11473  ege2le3  11608  cos01gt0  11699  sin02gt0  11700  cos12dec  11704  unennn  12326  dvef  13288  sin0pilem2  13303  cosq23lt0  13354  cosq34lt1  13371  cos02pilt1  13372  logbgcd1irraplemexp  13486  lgslem3  13503  trilpolemeq1  13879