let p, q, r, s be Element of LTLB_WFF ; :: thesis: for X being Subset of LTLB_WFF st X |- p => q & X |- r => s holds
X |- (p '&&' r) => (q '&&' s)
let X be Subset of LTLB_WFF; :: thesis: ( X |- p => q & X |- r => s implies X |- (p '&&' r) => (q '&&' s) )
assume that
A1: X |- p => q and
A2: X |- r => s ; :: thesis: X |- (p '&&' r) => (q '&&' s)
(p => q) => ((r => s) => ((p '&&' r) => (q '&&' s))) is ctaut by Th45;
then (p => q) => ((r => s) => ((p '&&' r) => (q '&&' s))) in LTL_axioms by LTLAXIO1:def 17;
then X |- (p => q) => ((r => s) => ((p '&&' r) => (q '&&' s))) by LTLAXIO1:42;
then X |- (r => s) => ((p '&&' r) => (q '&&' s)) by LTLAXIO1:43, A1;
hence X |- (p '&&' r) => (q '&&' s) by LTLAXIO1:43, A2; :: thesis: verum