Theorem 3brtr3i 4034

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 4028	. 2 ⊢ 𝐶𝑅𝐵
4		3brtr3.3	. 2 ⊢ 𝐵 = 𝐷
5	3, 4	breqtri 4030	1 ⊢ 𝐶𝑅𝐷

Syntax hints:   = wceq 1353   class class class wbr 4005

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006

This theorem is referenced by:  suplocsrlempr  7809  iap0  9145  ef01bndlem  11767