Theorem 3brtr3g 3961

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 3938	. 2 |- ( A R B <-> C R D )
5	1, 4	sylib 121	1 |- ( ph -> C R D )

Syntax hints:   -> wi 4   = wceq 1331   class class class wbr 3929

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 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930

This theorem is referenced by:  eqbrtrrid  3964  breqtrdi  3969  ssenen  6745  endjusym  6981  djuen  7067  ege2le3  11377