Theorem breqd 4095

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

Hypothesis

Ref	Expression
breq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
breqd	|- ( ph -> ( C A D <-> C B D ) )

Proof of Theorem breqd

Step	Hyp	Ref	Expression
1		breq1d.1	. 2 |- ( ph -> A = B )
2		breq 4086	. 2 |- ( A = B -> ( C A D <-> C B D ) )
3	1, 2	syl 14	1 |- ( ph -> ( C A D <-> C B D ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   class class class wbr 4084

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-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225  df-br 4085

This theorem is referenced by:  breq123d  4098  breqdi  4099  sbcbr12g  4140  supeq123d  7179  shftfibg  11368  shftfib  11371  2shfti  11379  prdsex  13339  prdsval  13343  eqgval  13797  dvdsrd  14095  unitpropdg  14149  znleval  14654  lmbr  14924  wlkpropg  16112  wlkv  16114  wlkvg  16116  trlsfvalg  16169  trlsv  16170