Could anyone help me on this problem? i think i did it wrong it looks wierd so i was hoping if someone could show me where i went wrong.

Prove the following using rules of inference.

(p -> q) ^ (r -> s) ^ [t -> ~(q V s) ^ t => (~p ^ ~r)

Set up

P -> q ... H1(Hypothesis 1)

r -> s ... H2(Hypothesis 2)

t -> ~(q V s) ... H3(Hypothesis 3)

t ... H4(Hypothesis 4)

Working:

t ^ [t -> ~(q V s)] <=> ~(q V s) ... H4 ^ H3 Modus ponens

<=> ~q ^ ~s (1) ... de morgan's law

p -> q => ~q -> ~p (2) ... Contrapositive

r -> s => ~s -> ~r (3) ... Contrapositive

(~q -> ~p) ^ (~q ^ ~s) .... (2) ^ (1)

=> ~p ^ ~s ... Exportation

=> ~s ^ ~p (4) ... Commutativity

(~s -> ~r) ^ (~s ^ ~p) (3) ^ (4)

=> ~r ^ ~p ... Exportation

=> ~p ^ ~r ... Commutativity <--- Conclusion

I might have the rules wrong, could some please check if i've done this correct? i really need to know its urgent. If i have done something wrong could you tell me where i went wrong and if possible could you state the correct rule of inference along with it