Theorem breqtrdi 4071

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 2193	. 2 |- A = A
3		breqtrdi.2	. 2 |- B = C
4	1, 2, 3	3brtr3g 4063	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1364   class class class wbr 4030

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031

This theorem is referenced by:  breqtrrdi  4072  en2eleq  7257  en2other2  7258  dju0en  7276  ltm1sr  7839  maxle2  11359  xrmax2sup  11400  mertenslem2  11682  ege2le3  11817  cos01gt0  11909  sin02gt0  11910  cos12dec  11914  unennn  12557  dvef  14906  sin0pilem2  14958  cosq23lt0  15009  cosq34lt1  15026  cos02pilt1  15027  logbgcd1irraplemexp  15141  lgslem3  15159  lgsquadlem1  15234  lgsquadlem3  15236  trilpolemeq1  15600