Theorem breqtrrd 3781

Description: Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)

Hypotheses

Ref	Expression
breqtrrd.1	|- (ph -> A RB)
breqtrrd.2	|- (ph -> C = B)

Assertion

Ref	Expression
breqtrrd	|- (ph -> A R C)

Proof of Theorem breqtrrd

Step	Hyp	Ref	Expression
1		breqtrrd.1	. 2 |- (ph -> A RB)
2		breqtrrd.2	. . 3 |- (ph -> C = B)
3	2	eqcomd 2042	. 2 |- (ph -> B = C)
4	1, 3	breqtrd 3779	1 |- (ph -> A R C)

Syntax hints:   -> wi 4   = wceq 1242   class class class wbr 3755

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756

This theorem is referenced by:  addlocprlemeq  6516  ltexprlemopl  6575  recexprlemloc  6603  cauappcvgprlemopl  6618  cauappcvgprlemladdfu  6626  cauappcvgprlem1  6631  caucvgprlemopl  6640  caucvgprlemladdfu  6648  mulgt0sr  6704  archsr  6708  mulap0r  7399  prodgt0  7599  uzsubsubfz  8681  fzctr  8761  leexp1a  8963  sqgt0ap  8975  sqge0  8983  nnlesq  9009  expnbnd  9025  cjmulge0  9117