Theorem ssbrd 4131

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 3226	. 2 |- ( ph -> ( <. C , D >. e. A -> <. C , D >. e. B ) )
3		df-br 4089	. 2 |- ( C A D <-> <. C , D >. e. A )
4		df-br 4089	. 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 2202   C_ wss 3200   <.cop 3672   class class class wbr 4088

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3206  df-ss 3213  df-br 4089

This theorem is referenced by:  ssbr  4132  ssbri  4133  sess1  4434  brrelex12  4764  coss1  4885  coss2  4886  eqbrrdva  4900  ersym  6713  ertr  6716  subrguss  14249  znleval  14666