let p, q be Element of HP-WFF ; :: thesis: for V being SetValuation
for P being Permutation of V
for p9 being Permutation of (SetVal V,p)
for q9 being Permutation of (SetVal V,q) st p9 = Perm P,p & q9 = Perm P,q holds
Perm P,(p => q) = p9 => q9
let V be SetValuation; :: thesis: for P being Permutation of V
for p9 being Permutation of (SetVal V,p)
for q9 being Permutation of (SetVal V,q) st p9 = Perm P,p & q9 = Perm P,q holds
Perm P,(p => q) = p9 => q9
let P be Permutation of V; :: thesis: for p9 being Permutation of (SetVal V,p)
for q9 being Permutation of (SetVal V,q) st p9 = Perm P,p & q9 = Perm P,q holds
Perm P,(p => q) = p9 => q9
A1: ex p9 being Permutation of (SetVal V,p) ex q9 being Permutation of (SetVal V,q) st
( p9 = (Perm P) . p & q9 = (Perm P) . q & (Perm P) . (p '&' q) = [:p9,q9:] & (Perm P) . (p => q) = p9 => q9 ) by Def5;
let p9 be Permutation of (SetVal V,p); :: thesis: for q9 being Permutation of (SetVal V,q) st p9 = Perm P,p & q9 = Perm P,q holds
Perm P,(p => q) = p9 => q9
let q9 be Permutation of (SetVal V,q); :: thesis: ( p9 = Perm P,p & q9 = Perm P,q implies Perm P,(p => q) = p9 => q9 )
assume ( p9 = Perm P,p & q9 = Perm P,q ) ; :: thesis: Perm P,(p => q) = p9 => q9
hence Perm P,(p => q) = p9 => q9 by A1; :: thesis: verum