Theorem 3brtr4i 5088

Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)

Hypotheses

Ref	Expression
3brtr4.1	⊢ 𝐴𝑅𝐵
3brtr4.2	⊢ 𝐶 = 𝐴
3brtr4.3	⊢ 𝐷 = 𝐵

Assertion

Ref	Expression
3brtr4i	⊢ 𝐶𝑅𝐷

Proof of Theorem 3brtr4i

Step	Hyp	Ref	Expression
1		3brtr4.2	. . 3 ⊢ 𝐶 = 𝐴
2		3brtr4.1	. . 3 ⊢ 𝐴𝑅𝐵
3	1, 2	eqbrtri 5079	. 2 ⊢ 𝐶𝑅𝐵
4		3brtr4.3	. 2 ⊢ 𝐷 = 𝐵
5	3, 4	breqtrri 5085	1 ⊢ 𝐶𝑅𝐷

Syntax hints:   = 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:  1lt2nq  10389  0lt1sr  10511  declt  12120  decltc  12121  decle  12126  fzennn  13330  faclbnd4lem1  13647  fsumabs  15150  ovolfiniun  24096  log2ublem3  25520  log2ub  25521  bclbnd  25850  bposlem8  25861  baseltedgf  26773  nmblolbii  28570  normlem6  28886  norm-ii-i  28908  nmbdoplbi  29795  dp2lt  30556  dp2ltsuc  30557  dp2ltc  30558  dplt  30575  dpltc  30578  dpmul4  30585  hgt750lemd  31914  hgt750lem  31917  supxrltinfxr  41717  nnsum4primesevenALTV  43960