let p, q be Element of CQC-WFF ; :: thesis: for x being bound_QC-variable holds (All x,(p <=> q)) => ((All x,p) <=> (All x,q)) is valid
let x be bound_QC-variable; :: thesis: (All x,(p <=> q)) => ((All x,p) <=> (All x,q)) is valid
A1: (All x,((p => q) '&' (q => p))) => ((p => q) '&' (q => p)) is valid by CQC_THE1:105;
(p <=> q) => (p <=> q) is valid ;
then (p <=> q) => ((p => q) '&' (q => p)) is valid by QC_LANG2:def 4;
then All x,((p <=> q) => ((p => q) '&' (q => p))) is valid by Th26;
then A2: (All x,(p <=> q)) => (All x,((p => q) '&' (q => p))) is valid by Th35;
( (All x,(p => q)) => ((All x,p) => (All x,q)) is valid & (All x,(q => p)) => ((All x,q) => (All x,p)) is valid ) by Th34;
then ((All x,(p => q)) '&' (All x,(q => p))) => (((All x,p) => (All x,q)) '&' ((All x,q) => (All x,p))) is valid by Lm5;
then A3: ((All x,(p => q)) '&' (All x,(q => p))) => ((All x,p) <=> (All x,q)) is valid by QC_LANG2:def 4;
A4: not x in still_not-bound_in (All x,((p => q) '&' (q => p))) by Th5;
((p => q) '&' (q => p)) => (q => p) is valid by Lm1;
then (All x,((p => q) '&' (q => p))) => (q => p) is valid by A1, LUKASI_1:43;
then A5: (All x,((p => q) '&' (q => p))) => (All x,(q => p)) is valid by A4, CQC_THE1:106;
((p => q) '&' (q => p)) => (p => q) is valid by Lm1;
then (All x,((p => q) '&' (q => p))) => (p => q) is valid by A1, LUKASI_1:43;
then (All x,((p => q) '&' (q => p))) => (All x,(p => q)) is valid by A4, CQC_THE1:106;
then (All x,((p => q) '&' (q => p))) => ((All x,(p => q)) '&' (All x,(q => p))) is valid by A5, Lm3;
then (All x,((p => q) '&' (q => p))) => ((All x,p) <=> (All x,q)) is valid by A3, LUKASI_1:43;
hence (All x,(p <=> q)) => ((All x,p) <=> (All x,q)) is valid by A2, LUKASI_1:43; :: thesis: verum