let Al be QC-alphabet ; :: thesis: for X being Subset of (CQC-WFF Al)

for p, q being Element of CQC-WFF Al st p in Cn X & p => q in Cn X holds

q in Cn X

let X be Subset of (CQC-WFF Al); :: thesis: for p, q being Element of CQC-WFF Al st p in Cn X & p => q in Cn X holds

q in Cn X

let p, q be Element of CQC-WFF Al; :: thesis: ( p in Cn X & p => q in Cn X implies q in Cn X )

assume A1: ( p in Cn X & p => q in Cn X ) ; :: thesis: q in Cn X

for T being Subset of (CQC-WFF Al) st T is being_a_theory & X c= T holds

q in T

for p, q being Element of CQC-WFF Al st p in Cn X & p => q in Cn X holds

q in Cn X

let X be Subset of (CQC-WFF Al); :: thesis: for p, q being Element of CQC-WFF Al st p in Cn X & p => q in Cn X holds

q in Cn X

let p, q be Element of CQC-WFF Al; :: thesis: ( p in Cn X & p => q in Cn X implies q in Cn X )

assume A1: ( p in Cn X & p => q in Cn X ) ; :: thesis: q in Cn X

for T being Subset of (CQC-WFF Al) st T is being_a_theory & X c= T holds

q in T

proof

hence
q in Cn X
by Def2; :: thesis: verum
let T be Subset of (CQC-WFF Al); :: thesis: ( T is being_a_theory & X c= T implies q in T )

assume that

A2: T is being_a_theory and

A3: X c= T ; :: thesis: q in T

( p in T & p => q in T ) by A1, A2, A3, Def2;

hence q in T by A2; :: thesis: verum

end;assume that

A2: T is being_a_theory and

A3: X c= T ; :: thesis: q in T

( p in T & p => q in T ) by A1, A2, A3, Def2;

hence q in T by A2; :: thesis: verum