Theorem breqtrdi 4171

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 4163	1 |- ( ph -> A R C )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   class class class wbr 4130

This proof depends on 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 proof 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 3715  df-pr 3716  df-op 3718  df-br 4131

This theorem is used by:  breqtrrdi  4172  en2eleq  7547  en2other2  7548  dju0en  7570  ltm1sr  8144  maxle2  11980  xrmax2sup  12022  mertenslem2  12305  ege2le3  12440  cos01gt0  12532  sin02gt0  12533  cos12dec  12537  bitsfzolem  12723  bitsmod  12725  unennn  13290  dvef  15830  sin0pilem2  15886  cosq23lt0  15937  cosq34lt1  15954  cos02pilt1  15955  logbgcd1irraplemexp  16076  pellexlem2  16098  lgslem3  16133  lgsquadlem1  16208  lgsquadlem3  16210  trilpolemeq1  17101