Theorem 3brtr3i 5114

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 5108	. 2 ⊢ 𝐶𝑅𝐵
4		3brtr3.3	. 2 ⊢ 𝐵 = 𝐷
5	3, 4	breqtri 5110	1 ⊢ 𝐶𝑅𝐷

Syntax hints:   = wceq 1542   class class class wbr 5085

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086

This theorem is referenced by:  supsrlem  11034  ef01bndlem  16151  pige3ALT  26484  log2ublem1  26910  log2ub  26913  ppiublem1  27165  logfacrlim2  27189  chebbnd1  27435  twocut  28415  bdayfinbndlem1  28459  nmoptri2i  32170  dpmul4  32973  problem5  35851  fouriersw  46659