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Metamath Proof Explorer

Theorem polcon2bN

Description: Contraposition law for polarity. (Contributed by NM, 23-Mar-2012) (New usage is discouraged.)

Ref Expression
Hypotheses 2polss.a 
|- A = ( Atoms ` K )
2polss.p 
|- ._|_ = ( _|_P ` K )
Assertion polcon2bN 
|- ( ( K e. HL /\ X C_ A /\ Y C_ A ) -> ( X C_ ( ._|_ ` Y ) <-> Y C_ ( ._|_ ` X ) ) )

Proof

Step Hyp Ref Expression
1 2polss.a 
 |-  A = ( Atoms ` K )
2 2polss.p 
 |-  ._|_ = ( _|_P ` K )
3 simpl1 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ X C_ ( ._|_ ` Y ) ) -> K e. HL )
4 simpl3 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ X C_ ( ._|_ ` Y ) ) -> Y C_ A )
5 simpr 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ X C_ ( ._|_ ` Y ) ) -> X C_ ( ._|_ ` Y ) )
6 1 2 polcon2N 
 |-  ( ( K e. HL /\ Y C_ A /\ X C_ ( ._|_ ` Y ) ) -> Y C_ ( ._|_ ` X ) )
7 3 4 5 6 syl3anc 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ X C_ ( ._|_ ` Y ) ) -> Y C_ ( ._|_ ` X ) )
8 simpl1 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ Y C_ ( ._|_ ` X ) ) -> K e. HL )
9 simpl2 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ Y C_ ( ._|_ ` X ) ) -> X C_ A )
10 simpr 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ Y C_ ( ._|_ ` X ) ) -> Y C_ ( ._|_ ` X ) )
11 1 2 polcon2N 
 |-  ( ( K e. HL /\ X C_ A /\ Y C_ ( ._|_ ` X ) ) -> X C_ ( ._|_ ` Y ) )
12 8 9 10 11 syl3anc 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ Y C_ ( ._|_ ` X ) ) -> X C_ ( ._|_ ` Y ) )
13 7 12 impbida 
 |-  ( ( K e. HL /\ X C_ A /\ Y C_ A ) -> ( X C_ ( ._|_ ` Y ) <-> Y C_ ( ._|_ ` X ) ) )