Theorem ssbri 4175

Description: Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)

Hypothesis

Ref	Expression
ssbri.1	|- A C_ B

Assertion

Ref	Expression
ssbri	|- ( C A D -> C B D )

Proof of Theorem ssbri

Step	Hyp	Ref	Expression
1		ssbri.1	. . . 4 |- A C_ B
2	1	a1i 9	. . 3 |- ( T. -> A C_ B )
3	2	ssbrd 4173	. 2 |- ( T. -> ( C A D -> C B D ) )
4	3	mptru 1411	1 |- ( C A D -> C B D )

This proof depends on syntax axioms:   -> wi 4   T. wtru 1403   C_ wss 3220   class class class wbr 4130

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233  df-br 4131

This theorem is used by:  brel  4827  swoer  6835  swoord1  6836  swoord2  6837  ecopover  6907  ecopoverg  6910  endom  7049