let A be non empty set ; :: thesis: for J being interpretation of A
for S being Element of CQC-Sub-WFF st ( for v being Element of Valuations_in A holds
( J,v |= CQC_Sub S iff J,v . (Val_S v,S) |= S ) ) holds
for v being Element of Valuations_in A holds
( J,v |= CQC_Sub (Sub_not S) iff J,v . (Val_S v,(Sub_not S)) |= Sub_not S )
let J be interpretation of A; :: thesis: for S being Element of CQC-Sub-WFF st ( for v being Element of Valuations_in A holds
( J,v |= CQC_Sub S iff J,v . (Val_S v,S) |= S ) ) holds
for v being Element of Valuations_in A holds
( J,v |= CQC_Sub (Sub_not S) iff J,v . (Val_S v,(Sub_not S)) |= Sub_not S )
let S be Element of CQC-Sub-WFF ; :: thesis: ( ( for v being Element of Valuations_in A holds
( J,v |= CQC_Sub S iff J,v . (Val_S v,S) |= S ) ) implies for v being Element of Valuations_in A holds
( J,v |= CQC_Sub (Sub_not S) iff J,v . (Val_S v,(Sub_not S)) |= Sub_not S ) )
assume A1: for v being Element of Valuations_in A holds
( J,v |= CQC_Sub S iff J,v . (Val_S v,S) |= S ) ; :: thesis: for v being Element of Valuations_in A holds
( J,v |= CQC_Sub (Sub_not S) iff J,v . (Val_S v,(Sub_not S)) |= Sub_not S )
let v be Element of Valuations_in A; :: thesis: ( J,v |= CQC_Sub (Sub_not S) iff J,v . (Val_S v,(Sub_not S)) |= Sub_not S )
A2: ( J,v |= 'not' (CQC_Sub S) iff not J,v |= CQC_Sub S ) by VALUAT_1:28;
( not J,v . (Val_S v,S) |= S iff J,v . (Val_S v,S) |= Sub_not S ) by Th18;
hence ( J,v |= CQC_Sub (Sub_not S) iff J,v . (Val_S v,(Sub_not S)) |= Sub_not S ) by A1, A2, Th19, SUBSTUT1:29; :: thesis: verum