Theorem 3brtr3i 4072

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 4066	. 2 ⊢ 𝐶𝑅𝐵
4		3brtr3.3	. 2 ⊢ 𝐵 = 𝐷
5	3, 4	breqtri 4068	1 ⊢ 𝐶𝑅𝐷

Syntax hints:   = wceq 1372   class class class wbr 4043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044

This theorem is referenced by:  suplocsrlempr  7919  iap0  9259  ef01bndlem  12038