let p, q, r be Element of LTLB_WFF ; :: thesis: for X being Subset of LTLB_WFF st X |- p => q & X |- p => r holds
X |- p => (q '&&' r)
let X be Subset of LTLB_WFF; :: thesis: ( X |- p => q & X |- p => r implies X |- p => (q '&&' r) )
assume that
A1: X |- p => q and
A2: X |- p => r ; :: thesis: X |- p => (q '&&' r)
set qr = q '&&' r;
(p => q) => ((p => r) => (p => (q '&&' r))) is ctaut by Th40;
then (p => q) => ((p => r) => (p => (q '&&' r))) in LTL_axioms by LTLAXIO1:def 17;
then X |- (p => q) => ((p => r) => (p => (q '&&' r))) by LTLAXIO1:42;
then X |- (p => r) => (p => (q '&&' r)) by LTLAXIO1:43, A1;
hence X |- p => (q '&&' r) by LTLAXIO1:43, A2; :: thesis: verum