Theorem 3brtr4i 5116

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 5107	. 2 ⊢ 𝐶𝑅𝐵
4		3brtr4.3	. 2 ⊢ 𝐷 = 𝐵
5	3, 4	breqtrri 5113	1 ⊢ 𝐶𝑅𝐷

Syntax hints:   = wceq 1542   class class class wbr 5086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087

This theorem is referenced by:  1lt2nq  10890  0lt1sr  11012  declt  12666  decltc  12667  decle  12672  fzennn  13924  faclbnd4lem1  14249  fsumabs  15758  basendxltplusgndx  17243  basendxlttsetndx  17312  basendxltplendx  17326  basendxltdsndx  17345  basendxltunifndx  17355  ovolfiniun  25481  log2ublem3  26928  log2ub  26929  bclbnd  27260  bposlem8  27271  basendxltedgfndx  29080  nmblolbii  30888  normlem6  31204  norm-ii-i  31226  nmbdoplbi  32113  dp2lt  32962  dp2ltsuc  32963  dp2ltc  32964  dplt  32981  dpltc  32984  dpmul4  32991  hgt750lemd  34811  hgt750lem  34814  supxrltinfxr  45898  nnsum4primesevenALTV  48292