Theorem breq123d 3881

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 3880	. 2 |- ( ph -> ( A R C <-> B R D ) )
4		breq123d.2	. . 3 |- ( ph -> R = S )
5	4	breqd 3878	. 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 1296   class class class wbr 3867

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-un 3017  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868

This theorem is referenced by:  sbcbrg  3916  fmptco  5503