Theorem ssbrd 4048

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 3156	. 2 |- ( ph -> ( <. C , D >. e. A -> <. C , D >. e. B ) )
3		df-br 4006	. 2 |- ( C A D <-> <. C , D >. e. A )
4		df-br 4006	. 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 2148   C_ wss 3131   <.cop 3597   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-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144  df-br 4006

This theorem is referenced by:  ssbri  4049  sess1  4339  brrelex12  4666  coss1  4784  coss2  4785  eqbrrdva  4799  ersym  6550  ertr  6553  subrguss  13363