Theorem breqtri 4030

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 4013	. 2 |- ( A R B <-> A R C )
4	1, 3	mpbi 145	1 |- A R C

Syntax hints:   = wceq 1353   class class class wbr 4005

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006

This theorem is referenced by:  breqtrri  4032  3brtr3i  4034  le9lt10  9413  9lt10  9517  sqrt2gt1lt2  11061  trireciplem  11511  cos1bnd  11770  cos2bnd  11771  cos01gt0  11773  sin4lt0  11777  z4even  11924  coseq00topi  14396  sincos4thpi  14401  lgsdir2lem2  14570  lgsdir2lem3  14571  ex-fl  14617