Theorem breqd 4016

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 4007	. 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 1353   class class class wbr 4005

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173  df-br 4006

This theorem is referenced by:  breq123d  4019  breqdi  4020  sbcbr12g  4060  supeq123d  6992  shftfibg  10831  shftfib  10834  2shfti  10842  prdsex  12723  eqgval  13087  dvdsrd  13268  unitpropdg  13322  lmbr  13752