Theorem ssbr 34008

Description: Subclass theorem for binary relation, in a more searchable form: ( R C_ S -> ( A R B -> A S B ) ). (Contributed by Peter Mazsa, 11-Nov-2019.)

Assertion

Ref	Expression
ssbr	|- ( A C_ B -> ( C A D -> C B D ) )

Proof of Theorem ssbr

Step	Hyp	Ref	Expression
1		ssel 3597	. 2 |- ( A C_ B -> ( <. C , D >. e. A -> <. C , D >. e. B ) )
2		df-br 4654	. 2 |- ( C A D <-> <. C , D >. e. A )
3		df-br 4654	. 2 |- ( C B D <-> <. C , D >. e. B )
4	1, 2, 3	3imtr4g 285	1 |- ( A C_ B -> ( C A D -> C B D ) )

Syntax hints:   -> wi 4   e. wcel 1990   C_ wss 3574   <.cop 4183   class class class wbr 4653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588  df-br 4654

This theorem is referenced by: (None)