Theorem 3brtr3g 4020

Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)

Hypotheses

Ref	Expression
3brtr3g.1	|- ( ph -> A R B )
3brtr3g.2	|- A = C
3brtr3g.3	|- B = D

Assertion

Ref	Expression
3brtr3g	|- ( ph -> C R D )

Proof of Theorem 3brtr3g

Step	Hyp	Ref	Expression
1		3brtr3g.1	. 2 |- ( ph -> A R B )
2		3brtr3g.2	. . 3 |- A = C
3		3brtr3g.3	. . 3 |- B = D
4	2, 3	breq12i 3996	. 2 |- ( A R B <-> C R D )
5	1, 4	sylib 121	1 |- ( ph -> C R D )

Syntax hints:   -> wi 4   = wceq 1348   class class class wbr 3987

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988

This theorem is referenced by:  eqbrtrrid  4023  breqtrdi  4028  ssenen  6826  endjusym  7070  djuen  7177  ege2le3  11623