let A be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) st X |- p => q holds
X \/ {p} |- q
let p, q be Element of CQC-WFF A; :: thesis: for X being Subset of (CQC-WFF A) st X |- p => q holds
X \/ {p} |- q
let X be Subset of (CQC-WFF A); :: thesis: ( X |- p => q implies X \/ {p} |- q )
p in {p} by TARSKI:def 1;
then p in X \/ {p} by XBOOLE_0:def 3;
then A1: X \/ {p} |- p by Th1;
assume X |- p => q ; :: thesis: X \/ {p} |- q
then X \/ {p} |- p => q by Th4, XBOOLE_1:7;
hence X \/ {p} |- q by A1, CQC_THE1:55; :: thesis: verum