Theorem breqtrdi 4169

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 2238	. 2 |- A = A
3		breqtrdi.2	. 2 |- B = C
4	1, 2, 3	3brtr3g 4161	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1402   class class class wbr 4128

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129

This theorem is referenced by:  breqtrrdi  4170  en2eleq  7541  en2other2  7542  dju0en  7564  ltm1sr  8138  maxle2  11961  xrmax2sup  12003  mertenslem2  12286  ege2le3  12421  cos01gt0  12513  sin02gt0  12514  cos12dec  12518  bitsfzolem  12704  bitsmod  12706  unennn  13271  dvef  15811  sin0pilem2  15866  cosq23lt0  15917  cosq34lt1  15934  cos02pilt1  15935  logbgcd1irraplemexp  16053  pellexlem2  16075  lgslem3  16104  lgsquadlem1  16179  lgsquadlem3  16181  trilpolemeq1  17063