Theorem 3brtr4g 4435

Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)

Hypotheses

Ref	Expression
3brtr4g.1	|- ( ph -> A R B )
3brtr4g.2	|- C = A
3brtr4g.3	|- D = B

Assertion

Ref	Expression
3brtr4g	|- ( ph -> C R D )

Proof of Theorem 3brtr4g

Step	Hyp	Ref	Expression
1		3brtr4g.1	. 2 |- ( ph -> A R B )
2		3brtr4g.2	. . 3 |- C = A
3		3brtr4g.3	. . 3 |- D = B
4	2, 3	breq12i 4411	. 2 |- ( C R D <-> A R B )
5	1, 4	sylibr 216	1 |- ( ph -> C R D )

Syntax hints:   -> wi 4   = wceq 1444   class class class wbr 4402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403

This theorem is referenced by:  syl5eqbr  4436  limensuci  7748  infensuc  7750  rlimneg  13710  isumsup2  13904  crt  14726  4sqlem6  14887  gzrngunit  19033  matgsum  19462  ovolunlem1a  22449  ovolfiniun  22454  ioombl1lem1  22511  ioombl1lem4  22514  iblss  22762  itgle  22767  dvfsumlem3  22980  emcllem6  23926  pntpbnd1a  24423  ostth2lem4  24474  omsmon  29126  omsmonOLD  29130  itg2gt0cn  31997  dalem-cly  33236  dalem10  33238  fourierdlem103  38073  fourierdlem104  38074