Theorem breq123d 3951

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 3950	. 2 |- ( ph -> ( A R C <-> B R D ) )
4		breq123d.2	. . 3 |- ( ph -> R = S )
5	4	breqd 3948	. 2 |- ( ph -> ( B R D <-> B S D ) )
6	3, 5	bitrd 187	1 |- ( ph -> ( A R C <-> B S D ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   class class class wbr 3937

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938

This theorem is referenced by:  sbcbrg  3990  fmptco  5594