Theorem breqtri 3810

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

Hypotheses

Ref	Expression
breqtr.1	|- A R B
breqtr.2	|- B = C

Assertion

Ref	Expression
breqtri	|- A R C

Proof of Theorem breqtri

Step	Hyp	Ref	Expression
1		breqtr.1	. 2 |- A R B
2		breqtr.2	. . 3 |- B = C
3	2	breq2i 3795	. 2 |- ( A R B <-> A R C )
4	1, 3	mpbi 143	1 |- A R C

Syntax hints:   = wceq 1285   class class class wbr 3787

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788

This theorem is referenced by:  breqtrri  3812  3brtr3i  3814  le9lt10  8573  9lt10  8677  sqrt2gt1lt2  10062  z4even  10449  ex-fl  10699