Theorem breqtrri 4142

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

Hypotheses

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

Assertion

Ref	Expression
breqtrri	|- A R C

Proof of Theorem breqtrri

Step	Hyp	Ref	Expression
1		breqtrr.1	. 2 |- A R B
2		breqtrr.2	. . 3 |- C = B
3	2	eqcomi 2238	. 2 |- B = C
4	1, 3	breqtri 4140	1 |- A R C

Syntax hints:   = wceq 1398   class class class wbr 4115

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 3701  df-pr 3702  df-op 3704  df-br 4116

This theorem is referenced by:  3brtr4i  4145  ensn1  7050  pw1dom2  7551  0lt1sr  8097  0le2  9348  2pos  9349  3pos  9352  4pos  9355  5pos  9358  6pos  9359  7pos  9360  8pos  9361  9pos  9362  1lt2  9428  2lt3  9429  3lt4  9431  4lt5  9434  5lt6  9438  6lt7  9443  7lt8  9449  8lt9  9456  nn0le2xi  9567  numltc  9756  declti  9768  sqge0i  11016  faclbnd2  11133  ege2le3  12387  cos2bnd  12476  3dvdsdec  12581  n2dvdsm1  12629  n2dvds3  12631  pockthi  13086  dec2dvds  13139  dveflem  15722  tangtx  15834  lgsdir2lem2  16033  ex-fl  16624