Formal Proof - Premises & Conclusion (Rules of Inference & Equivalence)

Question

So I have gathered/learned a total of 8 different rules of inference & 10 rules of equivalence for proofs: making a total of 18 proofs (Modus Ponens, Modus Tollens, Disjunctive Syllogism, Hypothetical Syllogism, Conjunction, Addition, Simplification, Constructive Dilemma, De Morgan's Law, Association, Distribution, Commutativity, Double Negation, Contraposition, Material Implication, Material Equivalence, Expotation, and Tautology). I want to turn the following premises GIVEN into a conclusion using the rules I know and mentioned.

Premises:

1. $(G \wedge I) \implies H$
2. $(I \implies H) \implies F$

Conclusion [What I want] : $G\implies F$

My Progress :

1. $(G \wedge I) \implies H$
2. $(I \implies H) \implies F\qquad\qquad\qquad\qquad [ G \implies F]$
3. $G \implies (I \implies H)\qquad\qquad\qquad\qquad [1, $exp]
4. $\sim(I \implies H) \vee F\qquad\qquad\qquad\qquad\quad\; [2, $Impl]

Not sure what else to do.

Answer 1

So I have gathered/learned a total of 8 different rules of inference & 10 rules of equivalence for proofs: making a total of 18 proofs (Modus Ponens, Modus Tollens, Disjunctive Syllogism, Hypothetical Syllogism, Conjunction, Addition, Simplification, Constructive Dilemma, De Morgan's Law, Association, Distribution, Commutativity, Double Negation, Contraposition, Material Implication, Material Equivalence, Expotation, and Tautology).

Hypothetical Syllogism is the rule of inference you seek:$$\dfrac{\phi\to\psi\qquad \psi\to\rho}{\phi\to\rho}$$

Answer 2

Using the first premise, we can find an equivalence : $$ (G \wedge I) \implies H \equiv (G\implies H)\vee (I\implies H) $$ We have that from second premise $(I \implies H) \implies F$ so notice that : $$ [(I \implies H)\wedge[(I \implies H) \implies F]]\implies F\qquad\text{[Modus Ponens]} $$ Therefore : $$ (G \wedge I) \implies H \equiv (G\implies F)\vee F$$ Which means $G\implies F$.