Theorem eqbrtri 4135

Description: Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
eqbrtr.1	⊢ 𝐴 = 𝐵
eqbrtr.2	⊢ 𝐵𝑅𝐶

Assertion

Ref	Expression
eqbrtri	⊢ 𝐴𝑅𝐶

Proof of Theorem eqbrtri

Step	Hyp	Ref	Expression
1		eqbrtr.2	. 2 ⊢ 𝐵𝑅𝐶
2		eqbrtr.1	. . 3 ⊢ 𝐴 = 𝐵
3	2	breq1i 4121	. 2 ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)
4	1, 3	mpbir 146	1 ⊢ 𝐴𝑅𝐶

Syntax hints:   = wceq 1398   class class class wbr 4114

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 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115

This theorem is referenced by:  eqbrtrri  4137  3brtr4i  4144  exmidpw2en  7185  exmidonfinlem  7509  neg1lt0  9365  halflt1  9475  3halfnz  9696  declei  9765  numlti  9766  faclbnd3  11133  geo2lim  12231  0.999...  12236  geoihalfsum  12237  fprodap0  12336  fprodap0f  12351  tan0  12446  cos2bnd  12475  sin4lt0  12482  eirraplem  12492  1nprm  12840  ballotfilemth  13229  znnen  13237  cnfldstr  14836  tan4thpi  15836  zabsle1  16002  ex-fl  16623  trilpolemisumle  16962