Theorem breqtrdi 4150

Description: A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)

Hypotheses

Ref	Expression
breqtrdi.1	⊢ (𝜑 → 𝐴𝑅𝐵)
breqtrdi.2	⊢ 𝐵 = 𝐶

Assertion

Ref	Expression
breqtrdi	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem breqtrdi

Step	Hyp	Ref	Expression
1		breqtrdi.1	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
2		eqid 2232	. 2 ⊢ 𝐴 = 𝐴
3		breqtrdi.2	. 2 ⊢ 𝐵 = 𝐶
4	1, 2, 3	3brtr3g 4142	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1398   class class class wbr 4109

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 2815  df-un 3215  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110

This theorem is referenced by:  breqtrrdi  4151  en2eleq  7498  en2other2  7499  dju0en  7521  ltm1sr  8092  maxle2  11897  xrmax2sup  11939  mertenslem2  12222  ege2le3  12357  cos01gt0  12449  sin02gt0  12450  cos12dec  12454  bitsfzolem  12640  bitsmod  12642  unennn  13148  dvef  15592  sin0pilem2  15647  cosq23lt0  15698  cosq34lt1  15715  cos02pilt1  15716  logbgcd1irraplemexp  15833  pellexlem2  15846  lgslem3  15875  lgsquadlem1  15950  lgsquadlem3  15952  trilpolemeq1  16824