Theorem ssbrd 4087

Description: Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)

Hypothesis

Ref	Expression
ssbrd.1	|- ( ph -> A C_ B )

Assertion

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

Proof of Theorem ssbrd

Step	Hyp	Ref	Expression
1		ssbrd.1	. . 3 |- ( ph -> A C_ B )
2	1	sseld 3192	. 2 |- ( ph -> ( <. C , D >. e. A -> <. C , D >. e. B ) )
3		df-br 4045	. 2 |- ( C A D <-> <. C , D >. e. A )
4		df-br 4045	. 2 |- ( C B D <-> <. C , D >. e. B )
5	2, 3, 4	3imtr4g 205	1 |- ( ph -> ( C A D -> C B D ) )

Syntax hints:   -> wi 4   e. wcel 2176   C_ wss 3166   <.cop 3636   class class class wbr 4044

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179  df-br 4045

This theorem is referenced by:  ssbri  4088  sess1  4384  brrelex12  4713  coss1  4833  coss2  4834  eqbrrdva  4848  ersym  6632  ertr  6635  subrguss  13998  znleval  14415