Theorem breqtrdi 4155

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 2234	. 2 |- A = A
3		breqtrdi.2	. 2 |- B = C
4	1, 2, 3	3brtr3g 4147	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1398   class class class wbr 4114

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115

This theorem is referenced by:  breqtrrdi  4156  en2eleq  7511  en2other2  7512  dju0en  7534  ltm1sr  8108  maxle2  11925  xrmax2sup  11967  mertenslem2  12250  ege2le3  12385  cos01gt0  12477  sin02gt0  12478  cos12dec  12482  bitsfzolem  12668  bitsmod  12670  unennn  13235  dvef  15721  sin0pilem2  15776  cosq23lt0  15827  cosq34lt1  15844  cos02pilt1  15845  logbgcd1irraplemexp  15962  pellexlem2  15975  lgslem3  16004  lgsquadlem1  16079  lgsquadlem3  16081  trilpolemeq1  16963