let p, q be Element of LTLB_WFF ; :: thesis: for F being Subset of LTLB_WFF st F |- p => q holds

F \/ {p} |- q

let F be Subset of LTLB_WFF; :: thesis: ( F |- p => q implies F \/ {p} |- q )

p in {p} by TARSKI:def 1;

then p in F \/ {p} by XBOOLE_0:def 3;

then A1: F \/ {p} |- p by Th42;

assume F |- p => q ; :: thesis: F \/ {p} |- q

then F \/ {p} |- p => q by Th56;

hence F \/ {p} |- q by A1, Th43; :: thesis: verum

F \/ {p} |- q

let F be Subset of LTLB_WFF; :: thesis: ( F |- p => q implies F \/ {p} |- q )

p in {p} by TARSKI:def 1;

then p in F \/ {p} by XBOOLE_0:def 3;

then A1: F \/ {p} |- p by Th42;

assume F |- p => q ; :: thesis: F \/ {p} |- q

then F \/ {p} |- p => q by Th56;

hence F \/ {p} |- q by A1, Th43; :: thesis: verum