Theorem breqtrdi 4005

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 2157	. 2 ⊢ 𝐴 = 𝐴
3		breqtrdi.2	. 2 ⊢ 𝐵 = 𝐶
4	1, 2, 3	3brtr3g 3997	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1335   class class class wbr 3965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966

This theorem is referenced by:  breqtrrdi  4006  en2eleq  7124  en2other2  7125  dju0en  7143  ltm1sr  7691  maxle2  11105  xrmax2sup  11144  mertenslem2  11426  ege2le3  11561  cos01gt0  11652  sin02gt0  11653  cos12dec  11657  unennn  12109  dvef  13059  sin0pilem2  13074  cosq23lt0  13125  cosq34lt1  13142  cos02pilt1  13143  logbgcd1irraplemexp  13256  trilpolemeq1  13582