#u abandali
#a h02

#q q04d

p <=> q, p & q <=> (p | q)

#check TP

p <=> q  <->  p & q <=> (p | q)

 1) p & q             <=> (p | q)
 2) (p & q   => p | q) &  (p | q => p & q)                by equiv
 3) (!(p & q) | p | q) & (!(p | q) | p & q)               by impl * 2
 4) (!p | !q  | p | q) & (!(p | q) | p & q)               by dm
 5) (true | !q | q)    & (!(p | q) | p & q)               by lem
 6) true               & (!(p | q) | p & q)               by simp1
 7)                       !(p | q) | p & q                by simp1
 8)                       !p & !q  | p & q                by dm
 9)                  (!p & !q | p) & (!p & !q | q)        by distr
10)            (!p | p) & (!q | p) & (!p | q) & (!q | q)  by distr * 2
11)                true & (!q | p) & (!p | q) & true      by lem * 2
12)                       (!q | p) & (!p | q)             by simp1 * 2
13)                       (q => p) & (p => q)             by impl * 2
14)                       p <=> q                         by equiv