Mastering Logical Equivalence Proofs: (P<->Q) ⊣ ⊢ ~(P<->~Q)

[Hazzardman]

~(P<->Q) ⊣ ⊢ (P<->~Q)

I'm suppose to write the proof for this equivalence but I can't figure it in either direction
The closest I got was (P->~Q) from ~(P<->Q) but I can't figure anything else out

 

[ZetaOfThree]

Do you mean to show that $\neg (P \leftrightarrow Q) \equiv (P \leftrightarrow \neg Q)$? Why not just use a truth table?

 

[Hazzardman]

I need to do it by formal proof
this is as far as I got but I can't figure out how to determine the necessay ~Q->P or how to do it in the opposite direction.

~(P<->Q)                want:P<->~Q
----------------------------------
|P                      want: ~Q
|-------------------------------
||Q                     reductio
||--------------------------------
|||P                    want: Q
|||--------------------------------
|||Q                    reiterate 3
||P->Q                  conditional introduction4-5
|||Q                    want: P
|||-------------------------------------------
|||P                    reiterate 2
||Q->P                  conditional introduction7-8
||P<->Q                 Biconditional definition 6,9
||~(P<>Q)               reiterate 1
|~Q                     indirect proof 3-11
P->~Q                   conditional introduction2-12

 

[Erland]

To answer, we need to know which axioms and rules of inference that are allowed in this context. This can differ in different textbooks.

 

[Hazzardman]

conjunction introduction
disjunction introduction
conjunction elimination
disjunction elimination
conditional elimination
biconditional elimination
negation introduction/elimination proof
conditional introduction proof
bicondional definition
reiteration

these are all the rules I have learned

 

[Stephen Tashi]

Unfortunately, the rules of logical inference don't all have standardized names. Their titles differ from textbook to textbook. Can you give a link to an article where those rules are written out?

 

[Erland]

I think I know what most of these rules are. But exactly how are negation introduction and elimination defined in your textbook?