Theorem 3brtr4i 5115

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 5106	. 2 ⊢ 𝐶𝑅𝐵
4		3brtr4.3	. 2 ⊢ 𝐷 = 𝐵
5	3, 4	breqtrri 5112	1 ⊢ 𝐶𝑅𝐷

Syntax hints:   = wceq 1542   class class class wbr 5085

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 2708

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 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086

This theorem is referenced by:  1lt2nq  10896  0lt1sr  11018  declt  12672  decltc  12673  decle  12678  fzennn  13930  faclbnd4lem1  14255  fsumabs  15764  basendxltplusgndx  17249  basendxlttsetndx  17318  basendxltplendx  17332  basendxltdsndx  17351  basendxltunifndx  17361  ovolfiniun  25468  log2ublem3  26912  log2ub  26913  bclbnd  27243  bposlem8  27254  basendxltedgfndx  29063  nmblolbii  30870  normlem6  31186  norm-ii-i  31208  nmbdoplbi  32095  dp2lt  32944  dp2ltsuc  32945  dp2ltc  32946  dplt  32963  dpltc  32966  dpmul4  32973  hgt750lemd  34792  hgt750lem  34795  supxrltinfxr  45877  nnsum4primesevenALTV  48277