Theorem ssbrd 4129

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 3224	. 2 |- ( ph -> ( <. C , D >. e. A -> <. C , D >. e. B ) )
3		df-br 4087	. 2 |- ( C A D <-> <. C , D >. e. A )
4		df-br 4087	. 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 2200   C_ wss 3198   <.cop 3670   class class class wbr 4086

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-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3204  df-ss 3211  df-br 4087

This theorem is referenced by:  ssbr  4130  ssbri  4131  sess1  4432  brrelex12  4762  coss1  4883  coss2  4884  eqbrrdva  4898  ersym  6709  ertr  6712  subrguss  14240  znleval  14657