Theorem breqtrdi 4075

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 2196	. 2 ⊢ 𝐴 = 𝐴
3		breqtrdi.2	. 2 ⊢ 𝐵 = 𝐶
4	1, 2, 3	3brtr3g 4067	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

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  7264  en2other2  7265  dju0en  7283  ltm1sr  7846  maxle2  11379  xrmax2sup  11421  mertenslem2  11703  ege2le3  11838  cos01gt0  11930  sin02gt0  11931  cos12dec  11935  bitsfzolem  12121  bitsmod  12123  unennn  12624  dvef  14973  sin0pilem2  15028  cosq23lt0  15079  cosq34lt1  15096  cos02pilt1  15097  logbgcd1irraplemexp  15214  lgslem3  15253  lgsquadlem1  15328  lgsquadlem3  15330  trilpolemeq1  15694