Theorem 3brtr3i 3952

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 3946	. 2 ⊢ 𝐶𝑅𝐵
4		3brtr3.3	. 2 ⊢ 𝐵 = 𝐷
5	3, 4	breqtri 3948	1 ⊢ 𝐶𝑅𝐷

Syntax hints:   = wceq 1331   class class class wbr 3924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925

This theorem is referenced by:  suplocsrlempr  7608  iap0  8936  ef01bndlem  11452