Theorem 3brtr4g 5200

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 5175	. 2 ⊢ (𝐶𝑅𝐷 ↔ 𝐴𝑅𝐵)
5	1, 4	sylibr 234	1 ⊢ (𝜑 → 𝐶𝑅𝐷)

Syntax hints:   → wi 4   = wceq 1537   class class class wbr 5166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167

This theorem is referenced by:  eqbrtrid  5201  enrefnn  9113  enpr2dOLD  9116  limensuci  9219  infensuc  9221  djuen  10239  djudom1  10252  rlimneg  15695  isumsup2  15894  crth  16825  4sqlem6  16990  gzrngunit  21474  matgsum  22464  ovolunlem1a  25550  ovolfiniun  25555  ioombl1lem1  25612  ioombl1lem4  25615  iblss  25860  itgle  25865  dvfsumlem3  26089  emcllem6  27062  gausslemma2dlem0f  27423  gausslemma2dlem0g  27424  pntpbnd1a  27647  ostth2lem4  27698  noinfbnd2lem1  27793  omsmon  34263  itg2gt0cn  37635  dalem-cly  39628  dalem10  39630  fourierdlem103  46130  fourierdlem104  46131