Theorem 3brtr4i 5119

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 5110	. 2 ⊢ 𝐶𝑅𝐵
4		3brtr4.3	. 2 ⊢ 𝐷 = 𝐵
5	3, 4	breqtrri 5116	1 ⊢ 𝐶𝑅𝐷

Syntax hints:   = wceq 1541   class class class wbr 5089

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090

This theorem is referenced by:  1lt2nq  10864  0lt1sr  10986  declt  12616  decltc  12617  decle  12622  fzennn  13875  faclbnd4lem1  14200  fsumabs  15708  basendxltplusgndx  17190  basendxlttsetndx  17259  basendxltplendx  17273  basendxltdsndx  17292  basendxltunifndx  17302  ovolfiniun  25429  log2ublem3  26885  log2ub  26886  bclbnd  27218  bposlem8  27229  basendxltedgfndx  28972  nmblolbii  30779  normlem6  31095  norm-ii-i  31117  nmbdoplbi  32004  dp2lt  32865  dp2ltsuc  32866  dp2ltc  32867  dplt  32884  dpltc  32887  dpmul4  32894  hgt750lemd  34661  hgt750lem  34664  supxrltinfxr  45557  nnsum4primesevenALTV  47911