1. A v B            Premise

2. A <--> (C & D)   Premise

3. B --> (D & G)    Premise

::4. A              Conditional Proof (CP) assumption #1

::5. (A --> (C & D)) & ((C & D) --> A)   2 Biconditional Equivalence (not sure of proper term)    

::6. (A --> (C & D))     5  Simplification (SIMP)

::7.  C & D              6,4 Modus Ponens (MP)

::8. D                   7  SIMP

::9. A --> D             4-8, CP 

::9. B                   CP assumption #2

::10. D & G              3,9  MP

::11. D                  10  SIMP

::12. B --> D            9-11 CP

::13. (A v B) --> D      4-12 CP Proof By Cases (PBC)

14. (A v B) --> D        4-13 CP (discharges CP assumptions)

15. D                    14,1  MP      



What does this proof do?  Lines 4-8 show if we assume A true, D is true (if A then D).  Lines 9-13 show "if B then D", therefore "if (A v B) then D" is true because we've covered the two possible cases (A true or B true) and we promote that result on line 14.  Line 15 follows from the premise on line 1.

 

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