Theorem 3brtr4g 5141

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

Hypotheses

Ref	Expression
3brtr4g.1	⊢ (𝜑 → 𝐴𝑅𝐵)
3brtr4g.2	⊢ 𝐶 = 𝐴
3brtr4g.3	⊢ 𝐷 = 𝐵

Assertion

Ref	Expression
3brtr4g	⊢ (𝜑 → 𝐶𝑅𝐷)

Proof of Theorem 3brtr4g

Step	Hyp	Ref	Expression
1		3brtr4g.1	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
2		3brtr4g.2	. . 3 ⊢ 𝐶 = 𝐴
3		3brtr4g.3	. . 3 ⊢ 𝐷 = 𝐵
4	2, 3	breq12i 5116	. 2 ⊢ (𝐶𝑅𝐷 ↔ 𝐴𝑅𝐵)
5	1, 4	sylibr 234	1 ⊢ (𝜑 → 𝐶𝑅𝐷)

Syntax hints:   → wi 4   = wceq 1540   class class class wbr 5107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108

This theorem is referenced by:  eqbrtrid  5142  enrefnn  9018  enpr2dOLD  9021  limensuci  9117  infensuc  9119  djuen  10123  djudom1  10136  rlimneg  15613  isumsup2  15812  crth  16748  4sqlem6  16914  gzrngunit  21350  matgsum  22324  ovolunlem1a  25397  ovolfiniun  25402  ioombl1lem1  25459  ioombl1lem4  25462  iblss  25706  itgle  25711  dvfsumlem3  25935  emcllem6  26911  gausslemma2dlem0f  27272  gausslemma2dlem0g  27273  pntpbnd1a  27496  ostth2lem4  27547  noinfbnd2lem1  27642  omsmon  34289  itg2gt0cn  37669  dalem-cly  39665  dalem10  39667  fourierdlem103  46207  fourierdlem104  46208