Theorem breqtrdi 4070

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 4062	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1364   class class class wbr 4029

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 3157  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030

This theorem is referenced by:  breqtrrdi  4071  en2eleq  7255  en2other2  7256  dju0en  7274  ltm1sr  7837  maxle2  11356  xrmax2sup  11397  mertenslem2  11679  ege2le3  11814  cos01gt0  11906  sin02gt0  11907  cos12dec  11911  unennn  12554  dvef  14873  sin0pilem2  14917  cosq23lt0  14968  cosq34lt1  14985  cos02pilt1  14986  logbgcd1irraplemexp  15100  lgslem3  15118  lgsquadlem1  15191  trilpolemeq1  15530