Theorem 3brtr4i 5196

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 5187	. 2 ⊢ 𝐶𝑅𝐵
4		3brtr4.3	. 2 ⊢ 𝐷 = 𝐵
5	3, 4	breqtrri 5193	1 ⊢ 𝐶𝑅𝐷

Syntax hints:   = wceq 1537   class class class wbr 5166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167

This theorem is referenced by:  1lt2nq  11042  0lt1sr  11164  declt  12786  decltc  12787  decle  12792  fzennn  14019  faclbnd4lem1  14342  fsumabs  15849  basendxltplusgndx  17340  basendxlttsetndx  17414  basendxltplendx  17428  basendxltdsndx  17447  basendxltunifndx  17457  ovolfiniun  25555  log2ublem3  27009  log2ub  27010  bclbnd  27342  bposlem8  27353  basendxltedgfndx  29028  baseltedgfOLD  29029  nmblolbii  30831  normlem6  31147  norm-ii-i  31169  nmbdoplbi  32056  dp2lt  32849  dp2ltsuc  32850  dp2ltc  32851  dplt  32868  dpltc  32871  dpmul4  32878  hgt750lemd  34625  hgt750lem  34628  supxrltinfxr  45364  nnsum4primesevenALTV  47675