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

proof

assume A1: p => q in TAUT ; :: thesis: p => ('not' ('not' q)) in TAUT
(p => q) => (p => ('not' ('not' q))) in TAUT by Th29;
hence p => ('not' ('not' q)) in TAUT by A1, CQC_THE1:82; :: thesis: verum

end;

assume A2: p => ('not' ('not' q)) in TAUT ; :: thesis: p => q in TAUT
(p => ('not' ('not' q))) => (p => q) in TAUT by Th29;
hence p => q in TAUT by A2, CQC_THE1:82; :: thesis: verum