Theorem breqtrdi 4030

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 2170	. 2 ⊢ 𝐴 = 𝐴
3		breqtrdi.2	. 2 ⊢ 𝐵 = 𝐶
4	1, 2, 3	3brtr3g 4022	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1348   class class class wbr 3989

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990

This theorem is referenced by:  breqtrrdi  4031  en2eleq  7172  en2other2  7173  dju0en  7191  ltm1sr  7739  maxle2  11176  xrmax2sup  11217  mertenslem2  11499  ege2le3  11634  cos01gt0  11725  sin02gt0  11726  cos12dec  11730  unennn  12352  dvef  13482  sin0pilem2  13497  cosq23lt0  13548  cosq34lt1  13565  cos02pilt1  13566  logbgcd1irraplemexp  13680  lgslem3  13697  trilpolemeq1  14072