Theorem 3brtr3i 5144

Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)

Hypotheses

Ref	Expression
3brtr3.1	⊢ 𝐴𝑅𝐵
3brtr3.2	⊢ 𝐴 = 𝐶
3brtr3.3	⊢ 𝐵 = 𝐷

Assertion

Ref	Expression
3brtr3i	⊢ 𝐶𝑅𝐷

Proof of Theorem 3brtr3i

Step	Hyp	Ref	Expression
1		3brtr3.2	. . 3 ⊢ 𝐴 = 𝐶
2		3brtr3.1	. . 3 ⊢ 𝐴𝑅𝐵
3	1, 2	eqbrtrri 5138	. 2 ⊢ 𝐶𝑅𝐵
4		3brtr3.3	. 2 ⊢ 𝐵 = 𝐷
5	3, 4	breqtri 5140	1 ⊢ 𝐶𝑅𝐷

Syntax hints:   = wceq 1567   class class class wbr 5113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114

This theorem is referenced by:  supsrlem  11096  ef01bndlem  16240  pige3ALT  26651  log2ublem1  27077  log2ub  27080  ppiublem1  27332  logfacrlim2  27356  chebbnd1  27602  twocut  28582  bdayfinbndlem1  28626  nmoptri2i  32392  dpmul4  33174  problem5  36094  fouriersw  46871