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Metamath Proof Explorer

Theorem chcon3i

Description: Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004) (New usage is discouraged.)

Ref Expression
Hypotheses ch0le.1 
|- A e. CH
chjcl.2 
|- B e. CH
Assertion chcon3i 
|- ( A = B <-> ( _|_ ` B ) = ( _|_ ` A ) )

Proof

Step Hyp Ref Expression
1 ch0le.1 
 |-  A e. CH
2 chjcl.2 
 |-  B e. CH
3 1 2 chsscon3i 
 |-  ( A C_ B <-> ( _|_ ` B ) C_ ( _|_ ` A ) )
4 2 1 chsscon3i 
 |-  ( B C_ A <-> ( _|_ ` A ) C_ ( _|_ ` B ) )
5 3 4 anbi12i 
 |-  ( ( A C_ B /\ B C_ A ) <-> ( ( _|_ ` B ) C_ ( _|_ ` A ) /\ ( _|_ ` A ) C_ ( _|_ ` B ) ) )
6 eqss 
 |-  ( A = B <-> ( A C_ B /\ B C_ A ) )
7 eqss 
 |-  ( ( _|_ ` B ) = ( _|_ ` A ) <-> ( ( _|_ ` B ) C_ ( _|_ ` A ) /\ ( _|_ ` A ) C_ ( _|_ ` B ) ) )
8 5 6 7 3bitr4i 
 |-  ( A = B <-> ( _|_ ` B ) = ( _|_ ` A ) )