Theorem 3brtr4i 5098

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 5089	. 2 ⊢ 𝐶𝑅𝐵
4		3brtr4.3	. 2 ⊢ 𝐷 = 𝐵
5	3, 4	breqtrri 5095	1 ⊢ 𝐶𝑅𝐷

Syntax hints:   = wceq 1537   class class class wbr 5068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069

This theorem is referenced by:  1lt2nq  10397  0lt1sr  10519  declt  12129  decltc  12130  decle  12135  fzennn  13339  faclbnd4lem1  13656  fsumabs  15158  ovolfiniun  24104  log2ublem3  25528  log2ub  25529  bclbnd  25858  bposlem8  25869  baseltedgf  26781  nmblolbii  28578  normlem6  28894  norm-ii-i  28916  nmbdoplbi  29803  dp2lt  30563  dp2ltsuc  30564  dp2ltc  30565  dplt  30582  dpltc  30585  dpmul4  30592  hgt750lemd  31921  hgt750lem  31924  supxrltinfxr  41731  nnsum4primesevenALTV  43973