# Some Practice Proofs Please Note: The rule "SI(X)" means that we would interpolate the proof of the previous result by the name of "X". P -> (Q <-> R) |- (P & Q) -> R 1 (1) P -> (Q <-> R) A 2 (2) P & Q A [goal: R] 2 (3) P 2 &E 1,2 (4) Q <-> R 1,3 ->E 1,2 (5) (Q -> R) & (R -> Q) 4 Df.<-> 1,2 (6) Q -> R 5 &E 2 (7) Q 2 &E 1,2 (8) R 6,7 ->E 1 (9) (P & Q) -> R 2,8 ->I P -> (Q v R) |- P -> (Q v S) 1 (1) P -> (Q v R) A 2 (2) R -> S A 3 (3) P A [goal: Q v S] 1,3 (4) Q v R 1,3 ->E 5 (5) Q A 5 (6) Q v S 5 vI 7 (7) R A 2,7 (8) S 2,7 ->E 2,7 (9) Q v S 8 vI 1,2,3 (10) Q v S 4,5,6,7,9 vE 1,2 (11) P -> (Q v S) 3,10 ->I (P v Q) & (R v S) |- ((P & R) v (P & S)) v ((Q & R) v (Q & S)) 1 (1) (P v Q) & (R v S) A 1 (2) P v Q 1 &E 1 (3) R v S 1 &E 4 (4) P A 5 (5) Q A 6 (6) R A 7 (7) S A 4,6 (8) P & R 4,6 &I 4,6 (9) (P & R) v (P & S) 8 vI 4,6 (10) ((P & R) v (P & S)) v ((Q & R) v (Q & S)) 9 vI 4,7 (11) P & S 4,7 &I 4,7 (12) (P & R) v (P & S) 11 vI 4,7 (13) ((P & R) v (P & S)) v ((Q & R) v (Q & S)) 12 vI 1,4 (14) ((P & R) v (P & S)) v ((Q & R) v (Q & S)) 3,6,10,7,13 vE 5,6 (15) Q & R 5,6 &I 5,6 (16) (Q & R) v (Q & S) 15 vI 5,6 (17) ((P & R) v (P & S)) v ((Q & R) v (Q & S)) 16 vI 5,7 (18) Q & S 4,7 &I 5,7 (19) (Q & R) v (Q & S) 18 vI 5,7 (20) ((P & R) v (P & S)) v ((Q & R) v (Q & S)) 19 vI 1,5 (21) ((P & R) v (P & S)) v ((Q & R) v (Q & S)) 3,6,10,7,13 vE 1 (22) ((P & R) v (P & S)) v ((Q & R) v (Q & S)) 2,4,14,5,21 vE (P -> -Q) -> (R v S) |- (P & Q) v (-S -> R) 1 (1) (P -> -Q) -> (R v S) A 1 (2) -(P -> -Q) v (R v S) 1 SI(Material Implication) 3 (3) -(P -> -Q) A 3 (4) P & --Q 3 SI(Material Implication) 3 (5) P 4 &E 3 (6) --Q 4 &E 3 (7) Q 6 -E 3 (8) P & Q 5,7 &I 3 (9) (P & Q) v (-S -> R) 8 vI 10 (10) R v S A 11 (11) R A 11 (12) --S v R 11 vI 13 (13) S A 13 (14) --S 13 -E 13 (15) --S v R 14 vI 10 (16) --S v R 10,11,12,13,15 vE 10 (17) -S -> R 16 SI(Material Implication) 10 (18) (P & Q) v (-S -> R) 17 vI 1 (19) (P & Q) v (-S -> R) 2,3,9,10,18 vE