Theorem breqtri 4156

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 4139	. 2 |- ( A R B <-> A R C )
4	1, 3	mpbi 145	1 |- A R C

This proof depends on syntax axioms:   = wceq 1402   class class class wbr 4131

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-br 4132

This theorem is used by:  breqtrri  4158  3brtr3i  4160  le9lt10  9819  9lt10  9923  sqrt2gt1lt2  11837  trireciplem  12292  cos1bnd  12551  cos2bnd  12552  cos01gt0  12555  sin4lt0  12559  z4even  12708  dec2dvds  13219  ballotfilem1c  13309  coseq00topi  16062  sincos4thpi  16067  log2ublem2  16222  log2ublem3  16223  birthdaylog2  16228  ppiublem1  16291  ppiqub  16293  bposlem4  16314  bposlem5  16315  bposlem9  16319  lgsdir2lem2  16353  lgsdir2lem3  16354  ex-fl  16944