Give a derivation of the following formal theorems:

Question

Show, φ, (φ $\Rightarrow$ ψ), (φ $\Rightarrow$ (ψ $\Rightarrow$ θ)) $\vdash$ θ

Here is my answer, please let me know if I'm doing this correctly!

1. φ $\qquad$$\qquad$$\qquad$< Hypothesis 1>
2. φ $\Rightarrow$ ψ $\qquad$$\qquad$ < Hypothesis 2>
3. φ $\Rightarrow$ (ψ $\Rightarrow$ θ)$\qquad$< Hypothesis 3>
4. ψ $\Rightarrow$ θ $\qquad$$\qquad$ <1,3 + Equanimity>
5. φ $\Rightarrow$ θ $\qquad$ $\qquad$<2,4 + Transitivity>
6. θ $\qquad$$\qquad$$\qquad$<1,5 + Equanimity>

Answer 1

Your proof is correct.

Well, the names are unfamiliar, but the steps are valid inferences.

"Equanimity" appears to be "Modus Ponens" and "Transitivity" is "Hypothetical Syllogism".

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You may also do it without "Transitivity", if you wish.

1. $φ\qquad \qquad \qquad~~$< Hypothesis 1>
2. $φ \to ψ \qquad\qquad$ < Hypothesis 2>
3. $φ \to (ψ \to θ)\quad~$< Hypothesis 3>
4. $ψ \qquad\qquad\qquad$ <1,2 + Equanimity>
5. $ψ \to θ \qquad\qquad$ <1,3 + Equanimity>
6. $θ \qquad\qquad\qquad$ <4,5 + Equanimity>