Theorem breqtri 4155

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 4138	. 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 4130

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 4131

This theorem is used by:  breqtrri  4157  3brtr3i  4159  le9lt10  9813  9lt10  9917  sqrt2gt1lt2  11831  trireciplem  12286  cos1bnd  12545  cos2bnd  12546  cos01gt0  12549  sin4lt0  12553  z4even  12702  dec2dvds  13213  ballotfilem1c  13303  coseq00topi  16028  sincos4thpi  16033  log2ublem2  16183  log2ublem3  16184  birthdaylog2  16189  ppiublem1  16252  ppiqub  16254  bposlem4  16275  bposlem5  16276  bposlem9  16280  lgsdir2lem2  16314  lgsdir2lem3  16315  ex-fl  16905