Theorem 3brtr4i 5173

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 5164	. 2 ⊢ 𝐶𝑅𝐵
4		3brtr4.3	. 2 ⊢ 𝐷 = 𝐵
5	3, 4	breqtrri 5170	1 ⊢ 𝐶𝑅𝐷

Syntax hints:   = wceq 1540   class class class wbr 5143

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144

This theorem is referenced by:  1lt2nq  11013  0lt1sr  11135  declt  12761  decltc  12762  decle  12767  fzennn  14009  faclbnd4lem1  14332  fsumabs  15837  basendxltplusgndx  17326  basendxlttsetndx  17399  basendxltplendx  17413  basendxltdsndx  17432  basendxltunifndx  17442  ovolfiniun  25536  log2ublem3  26991  log2ub  26992  bclbnd  27324  bposlem8  27335  basendxltedgfndx  29010  baseltedgfOLD  29011  nmblolbii  30818  normlem6  31134  norm-ii-i  31156  nmbdoplbi  32043  dp2lt  32867  dp2ltsuc  32868  dp2ltc  32869  dplt  32886  dpltc  32889  dpmul4  32896  hgt750lemd  34663  hgt750lem  34666  supxrltinfxr  45460  nnsum4primesevenALTV  47788