First of all, I know how to prove this theorem below:  

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> A statement form that contains $\leftrightarrow$ as its only connectives is a tautology iff each statement letter occurs an even number of times.
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To prove the theorem, I first derived this corollary:  

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> A statement built up using only $\leftrightarrow$ is true iff there are an even number of false terms.
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But now, I have to prove the theorem below:

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> Proving that a statement form that contains **$\lnot$ and** $\leftrightarrow$ as its only connectives is a tautology iff **$\lnot$ and** each statement letter occur an even number of times.
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The bold texts are the only differences comparing it to the previous question that I could solve. I tried modifying the corollary to help me prove the theorem but it didn't help me, or modifying the corollary made it impossible to prove. Is there anything else I could do?