let A be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF A holds
( p => q in TAUT A iff p => ('not' ('not' q)) in TAUT A )
let p, q be Element of CQC-WFF A; :: thesis: ( p => q in TAUT A iff p => ('not' ('not' q)) in TAUT A )
thus ( p => q in TAUT A implies p => ('not' ('not' q)) in TAUT A ) :: thesis: ( p => ('not' ('not' q)) in TAUT A implies p => q in TAUT A )

proof

assume A1: p => q in TAUT A ; :: thesis: p => ('not' ('not' q)) in TAUT A
(p => q) => (p => ('not' ('not' q))) in TAUT A by Th29;
hence p => ('not' ('not' q)) in TAUT A by A1, CQC_THE1:46; :: thesis: verum

end;

assume A2: p => ('not' ('not' q)) in TAUT A ; :: thesis: p => q in TAUT A
(p => ('not' ('not' q))) => (p => q) in TAUT A by Th29;
hence p => q in TAUT A by A2, CQC_THE1:46; :: thesis: verum