Question 1171518
<pre>
𝛼 → 𝛼 ⊨ 𝛽 → (𝛽 → 𝛼)

Substitute FALSITY for 𝛼, TRUTH for 𝛽, 
𝛼 → 𝛼 ⊨ 𝛽 → (𝛽 → 𝛼)
FALSITY → FALSITY ⊨ TRUTH → (TRUTH → FALSITY)
            TRUTH ⊨ TRUTH → FALSITY
            TRUTH ⊨ FALSITY


Does not hold.   



3.2.2. ¬(𝛽 → 𝛾) ⊨ 𝛾 ↔ 𝛼

Substitute FALSITY for 𝛼, TRUTH for 𝛽, and leave 𝛾 as it is,

    ¬(TRUTH → 𝛾) ⊨ 𝛾 ↔ FALSITY
  ¬(¬TRUTH or 𝛾) ⊨ 𝛾 ↔ FALSITY
 ¬(FALSITY or 𝛾) ⊨ 𝛾 ↔ FALSITY
 (¬FALSITY & ¬𝛾) ⊨ 𝛾 ↔ FALSITY
    (TRUTH & ¬𝛾) ⊨ 𝛾 ↔ FALSITY
              ¬𝛾 ⊨ 𝛾 ↔ FALSITY

Since the left side becomes ¬𝛾, then 𝛾 would have to be a falsity and
would imply falsity. 

Yes, it does hold.

Edwin</pre>