Theorem 3brtr4i 4873

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 4864	. 2 ⊢ 𝐶𝑅𝐵
4		3brtr4.3	. 2 ⊢ 𝐷 = 𝐵
5	3, 4	breqtrri 4870	1 ⊢ 𝐶𝑅𝐷

Syntax hints:   = wceq 1653   class class class wbr 4843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844

This theorem is referenced by:  1lt2nq  10083  0lt1sr  10204  declt  11812  decltc  11813  decle  11818  fzennn  13022  faclbnd4lem1  13333  fsumabs  14871  ovolfiniun  23609  log2ublem3  25027  log2ub  25028  emgt0  25085  bclbnd  25357  bposlem8  25368  baseltedgf  26229  nmblolbii  28179  normlem6  28497  norm-ii-i  28519  nmbdoplbi  29408  dp2lt  30109  dp2ltsuc  30110  dp2ltc  30111  dplt  30128  dpltc  30131  dpmul4  30138  hgt750lemd  31246  hgt750lem  31249  supxrltinfxr  40420  nnsum4primesevenALTV  42471