Theorem breqtrri 4136

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 2236	. 2 |- B = C
4	1, 3	breqtri 4134	1 |- A R C

Syntax hints:   = wceq 1398   class class class wbr 4109

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 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110

This theorem is referenced by:  3brtr4i  4139  ensn1  7036  pw1dom2  7537  0lt1sr  8080  0le2  9327  2pos  9328  3pos  9331  4pos  9334  5pos  9337  6pos  9338  7pos  9339  8pos  9340  9pos  9341  1lt2  9407  2lt3  9408  3lt4  9410  4lt5  9413  5lt6  9417  6lt7  9422  7lt8  9428  8lt9  9435  nn0le2xi  9546  numltc  9734  declti  9746  sqge0i  10988  faclbnd2  11104  ege2le3  12357  cos2bnd  12446  3dvdsdec  12551  n2dvdsm1  12599  n2dvds3  12601  pockthi  13056  dec2dvds  13109  dveflem  15591  tangtx  15703  lgsdir2lem2  15902  ex-fl  16493