Theorem 3brtr4i 5137

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 5128	. 2 ⊢ 𝐶𝑅𝐵
4		3brtr4.3	. 2 ⊢ 𝐷 = 𝐵
5	3, 4	breqtrri 5134	1 ⊢ 𝐶𝑅𝐷

Syntax hints:   = wceq 1540   class class class wbr 5107

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108

This theorem is referenced by:  1lt2nq  10926  0lt1sr  11048  declt  12677  decltc  12678  decle  12683  fzennn  13933  faclbnd4lem1  14258  fsumabs  15767  basendxltplusgndx  17249  basendxlttsetndx  17318  basendxltplendx  17332  basendxltdsndx  17351  basendxltunifndx  17361  ovolfiniun  25402  log2ublem3  26858  log2ub  26859  bclbnd  27191  bposlem8  27202  basendxltedgfndx  28921  nmblolbii  30728  normlem6  31044  norm-ii-i  31066  nmbdoplbi  31953  dp2lt  32805  dp2ltsuc  32806  dp2ltc  32807  dplt  32824  dpltc  32827  dpmul4  32834  hgt750lemd  34639  hgt750lem  34642  supxrltinfxr  45445  nnsum4primesevenALTV  47802