Theorem 3brtr4g 5092

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 5067	. 2 ⊢ (𝐶𝑅𝐷 ↔ 𝐴𝑅𝐵)
5	1, 4	sylibr 236	1 ⊢ (𝜑 → 𝐶𝑅𝐷)

Syntax hints:   → wi 4   = wceq 1533   class class class wbr 5058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059

This theorem is referenced by:  eqbrtrid  5093  enpr2d  8591  limensuci  8687  infensuc  8689  djuen  9589  djudom1  9602  rlimneg  14997  isumsup2  15195  crth  16109  4sqlem6  16273  gzrngunit  20605  matgsum  21040  ovolunlem1a  24091  ovolfiniun  24096  ioombl1lem1  24153  ioombl1lem4  24156  iblss  24399  itgle  24404  dvfsumlem3  24619  emcllem6  25572  gausslemma2dlem0f  25931  gausslemma2dlem0g  25932  pntpbnd1a  26155  ostth2lem4  26206  omsmon  31551  itg2gt0cn  34941  dalem-cly  36801  dalem10  36803  fourierdlem103  42488  fourierdlem104  42489