Theorem ssbrd 4032

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 3146	. 2 |- ( ph -> ( <. C , D >. e. A -> <. C , D >. e. B ) )
3		df-br 3990	. 2 |- ( C A D <-> <. C , D >. e. A )
4		df-br 3990	. 2 |- ( C B D <-> <. C , D >. e. B )
5	2, 3, 4	3imtr4g 204	1 |- ( ph -> ( C A D -> C B D ) )

Syntax hints:   -> wi 4   e. wcel 2141   C_ wss 3121   <.cop 3586   class class class wbr 3989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134  df-br 3990

This theorem is referenced by:  ssbri  4033  sess1  4322  brrelex12  4649  coss1  4766  coss2  4767  eqbrrdva  4781  ersym  6525  ertr  6528