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I want to derive the following:

∀x(Fx ↔ (¬Gx ∨ ¬Hx)). ¬∀x(Gx ∧ Hx) → ∃x(Ix ∧ ¬Gx) ∴ ∃xFx → ∃x(Ix ∧ Fx)

This is my attempt:

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Any suggestions as to how I continue and derive this? I cannot figure out how to continue
Thank you

EDIT: These are the inference rules as well as the derivation rules and assumptions:

enter image description here

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  • Could you clarify what rules of inference are you using and include the proof checker website ?
    – F. Zer
    Apr 22 '20 at 11:59
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    @F.Zer I have edited it to show above. The proof checker is called Elogic
    – applepi
    Apr 22 '20 at 12:05
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I do not have Elogic software, but perhaps this gives you a rough guide about the proof. I used some of @Mauro Allegranza excellent suggestions.

You assume ∃xFx and then assume ¬∃x(Ix ∧ Fx) in order to use Indirect Proof (your system calls this rule Indirect Derivation).

enter image description here

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  • Thank you. Can I ask do you know where I can find a guide to the definitions you have used? @F.Zer
    – applepi
    Apr 22 '20 at 13:59
  • You are welcome. Here proofs.openlogicproject.org is the proof checker I used in the proof. In the right side of this webpage, there is a reference of the rules used.
    – F. Zer
    Apr 22 '20 at 14:37

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