Hello every one!

It is about a propositional logic problem, I have tried it vey hard maybe... cause I am too tired...

This is it: a^b-> c |- (a->c)^(b->c) which means that (a->c)^(b->c) is a Logical consequence of a^b-> c. Also means that this trunk of tree:

a^b-> c -requirement

~[(a->c)^(b->c) ] -negative consequence

should give an open tree because {a=t,b=c=f} and {a=f, b=t, c=f} make true this:

((a^b) -> c)^~((a->c)^(b->c)).

This is the tree:

a^b->c(1)

~((a->c)^(b->c))(2)

---------------------

(3) ~(a^b) c from (1)

--------------------

(4) ~(a->c) (5)~(b->c) from (2)

----------------------

~a ~b from (3)

-----------------------------------------

a

~c from (4)

-------------------------------------

b

~c from (5)

------------------------------------------

this tree has all branches closed. Why is this, what I am doing wrong? Any help, any minor idea would be very appreciated!

Thank you all in advance,

Melsi