Theorem 3brtr4i 5102

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 5093	. 2 ⊢ 𝐶𝑅𝐵
4		3brtr4.3	. 2 ⊢ 𝐷 = 𝐵
5	3, 4	breqtrri 5099	1 ⊢ 𝐶𝑅𝐷

Syntax hints:   = wceq 1547   class class class wbr 5072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073

This theorem is referenced by:  1lt2nq  10887  0lt1sr  11009  declt  12663  decltc  12664  decle  12669  fzennn  13921  faclbnd4lem1  14246  fsumabs  15755  basendxltplusgndx  17240  basendxlttsetndx  17309  basendxltplendx  17323  basendxltdsndx  17342  basendxltunifndx  17352  ovolfiniun  25486  log2ublem3  26930  log2ub  26931  bclbnd  27261  bposlem8  27272  basendxltedgfndx  29081  nmblolbii  30888  normlem6  31204  norm-ii-i  31226  nmbdoplbi  32113  dp2lt  32963  dp2ltsuc  32964  dp2ltc  32965  dplt  32982  dpltc  32985  dpmul4  32992  hgt750lemd  34832  hgt750lem  34835  supxrltinfxr  45892  nnsum4primesevenALTV  48292