let A be QC-alphabet ; :: thesis: for p, q, r being Element of CQC-WFF A st p => q in TAUT A & q => r in TAUT A holds

p => r in TAUT A

let p, q, r be Element of CQC-WFF A; :: thesis: ( p => q in TAUT A & q => r in TAUT A implies p => r in TAUT A )

assume that

A1: p => q in TAUT A and

A2: q => r in TAUT A ; :: thesis: p => r in TAUT A

(p => q) => ((q => r) => (p => r)) in TAUT A by Th1;

then (q => r) => (p => r) in TAUT A by A1, CQC_THE1:46;

hence p => r in TAUT A by A2, CQC_THE1:46; :: thesis: verum

p => r in TAUT A

let p, q, r be Element of CQC-WFF A; :: thesis: ( p => q in TAUT A & q => r in TAUT A implies p => r in TAUT A )

assume that

A1: p => q in TAUT A and

A2: q => r in TAUT A ; :: thesis: p => r in TAUT A

(p => q) => ((q => r) => (p => r)) in TAUT A by Th1;

then (q => r) => (p => r) in TAUT A by A1, CQC_THE1:46;

hence p => r in TAUT A by A2, CQC_THE1:46; :: thesis: verum