Theorem breqtrdi 4039

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 2175	. 2 |- A = A
3		breqtrdi.2	. 2 |- B = C
4	1, 2, 3	3brtr3g 4031	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1353   class class class wbr 3998

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999

This theorem is referenced by:  breqtrrdi  4040  en2eleq  7184  en2other2  7185  dju0en  7203  ltm1sr  7751  maxle2  11187  xrmax2sup  11228  mertenslem2  11510  ege2le3  11645  cos01gt0  11736  sin02gt0  11737  cos12dec  11741  unennn  12363  dvef  13739  sin0pilem2  13754  cosq23lt0  13805  cosq34lt1  13822  cos02pilt1  13823  logbgcd1irraplemexp  13937  lgslem3  13954  trilpolemeq1  14329