Question 1205226
<pre>
1.  (P --> Q) ==  P --> (Q v ~P) 

// Note to prove A == B, need to show A --> B and  B --> A
// Also note I'm using simplified notation as it is easier to enter

2. :: P                                Conditional Proof (CP) assumption #1
3. :: Q                                2,1   Modus Ponens (MP)
4. :: Q v ~P                           3     Addition (ADD) 
5. :: P --> (Q v ~P)                   2-4  CP
6. :: (P --> Q) --> (P --> (Q v ~P))   1-5  CP

// Now go the other way
7. :: P --> (Q v ~P)                   CP assumption #2
8. :: P                                CP assumption #3
9. :: (Q v ~P)                         8,7 MP
10.:: P --> Q                          9   Material Implication (MI)
11.:: (P --> (Q v ~P)) -->  (P --> Q)  7-10  CP
12.:: (P --> Q) == (P --> (Q v ~P))    6,11  Material Equivalence (ME)
13.(P --> Q) == (P --> (Q v ~P))       2-12  CP

There may be shorter ways to go, but this is what came to mind.