Theorem breqtrdi 4152

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 4144	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1398   class class class wbr 4111

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 3217  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112

This theorem is referenced by:  breqtrrdi  4153  en2eleq  7500  en2other2  7501  dju0en  7523  ltm1sr  8097  maxle2  11905  xrmax2sup  11947  mertenslem2  12230  ege2le3  12365  cos01gt0  12457  sin02gt0  12458  cos12dec  12462  bitsfzolem  12648  bitsmod  12650  unennn  13169  dvef  15641  sin0pilem2  15696  cosq23lt0  15747  cosq34lt1  15764  cos02pilt1  15765  logbgcd1irraplemexp  15882  pellexlem2  15895  lgslem3  15924  lgsquadlem1  15999  lgsquadlem3  16001  trilpolemeq1  16873