Theorem breqtrdi 4075

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 2196	. 2 |- A = A
3		breqtrdi.2	. 2 |- B = C
4	1, 2, 3	3brtr3g 4067	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1364   class class class wbr 4034

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035

This theorem is referenced by:  breqtrrdi  4076  en2eleq  7274  en2other2  7275  dju0en  7297  ltm1sr  7861  maxle2  11394  xrmax2sup  11436  mertenslem2  11718  ege2le3  11853  cos01gt0  11945  sin02gt0  11946  cos12dec  11950  bitsfzolem  12136  bitsmod  12138  unennn  12639  dvef  15047  sin0pilem2  15102  cosq23lt0  15153  cosq34lt1  15170  cos02pilt1  15171  logbgcd1irraplemexp  15288  lgslem3  15327  lgsquadlem1  15402  lgsquadlem3  15404  trilpolemeq1  15771