let Al be QC-alphabet ; :: thesis: for A being non empty set
for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al st ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) ) holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (Sub_not S) iff J,v . (Val_S (v,(Sub_not S))) |= Sub_not S )
let A be non empty set ; :: thesis: for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al st ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) ) holds
for v being Element of Valuations_in (Al,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 Al,A; :: thesis: for S being Element of CQC-Sub-WFF Al st ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) ) holds
for v being Element of Valuations_in (Al,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 Al; :: thesis: ( ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) ) implies for v being Element of Valuations_in (Al,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 (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) ; :: thesis: for v being Element of Valuations_in (Al,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 (Al,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_1:17;
( not J,v . (Val_S (v,S)) |= S iff J,v . (Val_S (v,S)) |= Sub_not S ) by Th17;
hence ( J,v |= CQC_Sub (Sub_not S) iff J,v . (Val_S (v,(Sub_not S))) |= Sub_not S ) by A1, A2, Th18, SUBSTUT1:29; :: thesis: verum