Theorem nelcon3d 2909

Description: Contrapositive law deduction for negated membership. (Contributed by AV, 28-Jan-2020.)

Hypothesis

Ref	Expression
nelcon3d.1	|- ( ph -> ( A e. B -> C e. D ) )

Assertion

Ref	Expression
nelcon3d	|- ( ph -> ( C e/ D -> A e/ B ) )

Proof of Theorem nelcon3d

Step	Hyp	Ref	Expression
1		nelcon3d.1	. . 3 |- ( ph -> ( A e. B -> C e. D ) )
2	1	con3d 148	. 2 |- ( ph -> ( -. C e. D -> -. A e. B ) )
3		df-nel 2898	. 2 |- ( C e/ D <-> -. C e. D )
4		df-nel 2898	. 2 |- ( A e/ B <-> -. A e. B )
5	2, 3, 4	3imtr4g 285	1 |- ( ph -> ( C e/ D -> A e/ B ) )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1990   e/ wnel 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-nel 2898

This theorem is referenced by:  prcssprc  4806  lcmfnnval  15337  isnmgm  17246  mgmplusfreseq  41773