Theorem 3brtr4g 4120

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

Hypotheses

Ref	Expression
3brtr4g.1	|- ( ph -> A R B )
3brtr4g.2	|- C = A
3brtr4g.3	|- D = B

Assertion

Ref	Expression
3brtr4g	|- ( ph -> C R D )

Proof of Theorem 3brtr4g

Step	Hyp	Ref	Expression
1		3brtr4g.1	. 2 |- ( ph -> A R B )
2		3brtr4g.2	. . 3 |- C = A
3		3brtr4g.3	. . 3 |- D = B
4	2, 3	breq12i 4095	. 2 |- ( C R D <-> A R B )
5	1, 4	sylibr 134	1 |- ( ph -> C R D )

Syntax hints:   -> wi 4   = wceq 1395   class class class wbr 4086

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087

This theorem is referenced by:  eqbrtrid  4121  enpr2d  6992  crth  12786  4sqlem6  12946  gausslemma2dlem0f  15773  gausslemma2dlem0g  15774  wlkcompim  16149  upgrwlkcompim  16159  trilpolemgt1  16579