Theorem 3brtr4g 5143

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 237	1 ⊢ (𝜑 → 𝐶𝑅𝐷)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   class class class wbr 5107

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108

This theorem is used by:  eqbrtrid  5144  enrefnn  9056  limensuci  9154  infensuc  9156  djuen  10175  djudom1  10188  rlimneg  15736  isumsup2  15937  crth  16873  4sqlem6  17039  gzrngunit  21647  matgsum  22660  ovolunlem1a  25725  ovolfiniun  25730  ioombl1lem1  25787  ioombl1lem4  25790  iblss  26034  itgle  26039  dvfsumlem3  26257  emcllem6  27235  gausslemma2dlem0f  27595  gausslemma2dlem0g  27596  pntpbnd1a  27819  ostth2lem4  27870  noinfbnd2lem1  27964  omsmon  34796  itg2gt0cn  38411  dalem-cly  40531  dalem10  40533  fourierdlem103  47024  fourierdlem104  47025