Theorem 3brtr4i 5139

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 5130	. 2 ⊢ 𝐶𝑅𝐵
4		3brtr4.3	. 2 ⊢ 𝐷 = 𝐵
5	3, 4	breqtrri 5136	1 ⊢ 𝐶𝑅𝐷

This proof depends on syntax axioms:   = wceq 1570   class class class wbr 5107

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108

This theorem is used by:  1lt2nq  10986  0lt1sr  11108  declt  12773  decltc  12774  decle  12779  fzennn  14036  faclbnd4lem1  14361  fsumabs  15892  basendxltplusgndx  17377  basendxlttsetndx  17446  basendxltplendx  17460  basendxltdsndx  17479  basendxltunifndx  17489  ovolfiniun  25735  log2ublem3  27193  log2ub  27194  bclbnd  27524  bposlem8  27535  basendxltedgfndx  29459  nmblolbii  31288  normlem6  31604  norm-ii-i  31626  nmbdoplbi  32513  dp2lt  33338  dp2ltsuc  33339  dp2ltc  33340  dplt  33357  dpltc  33360  dpmul4  33367  hgt750lemd  35164  hgt750lem  35167  supxrltinfxr  46285  goldratval  47762  nnsum4primesevenALTV  48725