let p, q, r be Element of LTLB_WFF ; :: thesis: for X being Subset of LTLB_WFF st X |- p => q & X |- p => r & X |- r => p holds
X |- r => q
let X be Subset of LTLB_WFF; :: thesis: ( X |- p => q & X |- p => r & X |- r => p implies X |- r => q )
assume that
A1: X |- p => q and
A2: X |- p => r and
A3: X |- r => p ; :: thesis: X |- r => q
(p => q) => ((p => r) => ((r => p) => (r => q))) is ctaut by Th46;
then (p => q) => ((p => r) => ((r => p) => (r => q))) in LTL_axioms by LTLAXIO1:def 17;
then X |- (p => q) => ((p => r) => ((r => p) => (r => q))) by LTLAXIO1:42;
then X |- (p => r) => ((r => p) => (r => q)) by LTLAXIO1:43, A1;
then X |- (r => p) => (r => q) by LTLAXIO1:43, A2;
hence X |- r => q by LTLAXIO1:43, A3; :: thesis: verum