Theorem breqtrdi 4156

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 2234	. 2 |- A = A
3		breqtrdi.2	. 2 |- B = C
4	1, 2, 3	3brtr3g 4148	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1398   class class class wbr 4115

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-sn 3701  df-pr 3702  df-op 3704  df-br 4116

This theorem is referenced by:  breqtrrdi  4157  en2eleq  7513  en2other2  7514  dju0en  7536  ltm1sr  8110  maxle2  11928  xrmax2sup  11970  mertenslem2  12253  ege2le3  12388  cos01gt0  12480  sin02gt0  12481  cos12dec  12485  bitsfzolem  12671  bitsmod  12673  unennn  13238  dvef  15723  sin0pilem2  15778  cosq23lt0  15829  cosq34lt1  15846  cos02pilt1  15847  logbgcd1irraplemexp  15964  pellexlem2  15977  lgslem3  16006  lgsquadlem1  16081  lgsquadlem3  16083  trilpolemeq1  16965