Theorem 3brtr4i 5060

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 5051	. 2 ⊢ 𝐶𝑅𝐵
4		3brtr4.3	. 2 ⊢ 𝐷 = 𝐵
5	3, 4	breqtrri 5057	1 ⊢ 𝐶𝑅𝐷

Syntax hints:   = wceq 1538   class class class wbr 5030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031

This theorem is referenced by:  1lt2nq  10384  0lt1sr  10506  declt  12114  decltc  12115  decle  12120  fzennn  13331  faclbnd4lem1  13649  fsumabs  15148  ovolfiniun  24105  log2ublem3  25534  log2ub  25535  bclbnd  25864  bposlem8  25875  baseltedgf  26787  nmblolbii  28582  normlem6  28898  norm-ii-i  28920  nmbdoplbi  29807  dp2lt  30587  dp2ltsuc  30588  dp2ltc  30589  dplt  30606  dpltc  30609  dpmul4  30616  hgt750lemd  32029  hgt750lem  32032  3lexlogpow5ineq2  39342  supxrltinfxr  42087  nnsum4primesevenALTV  44319