Theorem breqtrdi 4129

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 2231	. 2 |- A = A
3		breqtrdi.2	. 2 |- B = C
4	1, 2, 3	3brtr3g 4121	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1397   class class class wbr 4088

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089

This theorem is referenced by:  breqtrrdi  4130  en2eleq  7406  en2other2  7407  dju0en  7429  ltm1sr  7997  maxle2  11773  xrmax2sup  11815  mertenslem2  12098  ege2le3  12233  cos01gt0  12325  sin02gt0  12326  cos12dec  12330  bitsfzolem  12516  bitsmod  12518  unennn  13019  dvef  15453  sin0pilem2  15508  cosq23lt0  15559  cosq34lt1  15576  cos02pilt1  15577  logbgcd1irraplemexp  15694  lgslem3  15733  lgsquadlem1  15808  lgsquadlem3  15810  trilpolemeq1  16647