Theorem breq123d 4116

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 4115	. 2 |- ( ph -> ( A R C <-> B R D ) )
4		breq123d.2	. . 3 |- ( ph -> R = S )
5	4	breqd 4113	. 2 |- ( ph -> ( B R D <-> B S D ) )
6	3, 5	bitrd 188	1 |- ( ph -> ( A R C <-> B S D ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   class class class wbr 4102

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-un 3214  df-sn 3688  df-pr 3689  df-op 3691  df-br 4103

This theorem is referenced by:  sbcbrg  4157  fmptco  5834