Theorem breqtrdi 4149

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 2232	. 2 |- A = A
3		breqtrdi.2	. 2 |- B = C
4	1, 2, 3	3brtr3g 4141	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1398   class class class wbr 4108

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 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-un 3214  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109

This theorem is referenced by:  breqtrrdi  4150  en2eleq  7497  en2other2  7498  dju0en  7520  ltm1sr  8088  maxle2  11890  xrmax2sup  11932  mertenslem2  12215  ege2le3  12350  cos01gt0  12442  sin02gt0  12443  cos12dec  12447  bitsfzolem  12633  bitsmod  12635  unennn  13137  dvef  15579  sin0pilem2  15634  cosq23lt0  15685  cosq34lt1  15702  cos02pilt1  15703  logbgcd1irraplemexp  15820  pellexlem2  15833  lgslem3  15862  lgsquadlem1  15937  lgsquadlem3  15939  trilpolemeq1  16811