Theorem breq123d 3799

Description: Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)

Hypotheses

Ref	Expression
breq1d.1	|- ( ph -> A = B )
breq123d.2	|- ( ph -> R = S )
breq123d.3	|- ( ph -> C = D )

Assertion

Ref	Expression
breq123d	|- ( ph -> ( A R C <-> B S D ) )

Proof of Theorem breq123d

Step	Hyp	Ref	Expression
1		breq1d.1	. . 3 |- ( ph -> A = B )
2		breq123d.3	. . 3 |- ( ph -> C = D )
3	1, 2	breq12d 3798	. 2 |- ( ph -> ( A R C <-> B R D ) )
4		breq123d.2	. . 3 |- ( ph -> R = S )
5	4	breqd 3796	. 2 |- ( ph -> ( B R D <-> B S D ) )
6	3, 5	bitrd 186	1 |- ( ph -> ( A R C <-> B S D ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   class class class wbr 3785

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786

This theorem is referenced by:  sbcbrg  3834  fmptco  5351