This question is from “Introduction to Mathematical Logic” by Elliot Mendelson , forth edition , page 75.

In page 75 of the book , there is a corollary in a paragraph that can be used to apply the **Deduction Theorem** multiple times in a row.

The new proof of $Gamma vdash mathscr B to mathscr C$ (in Proposition 2.4 $Gamma vdash mathscr C$ )

involves an application of Gen to a wf depending upon a wf $mathscr E$ of $Gamma$ only if

there is an application of Gen in the given proof of $Gamma, mathscr B vdash mathscr C$ that involves

the same quantified variable and is applied to a wf that depends upon $mathscr E$.

But there seems to be no justification/proof about this corollary.I searched online but non of them seem to mention this corollary.Are there any references about this I can read or is it just too trivial?

Here is an failed attempt of mine for justifying this corollary.

1.Technique for applying deduction multiple times in Mendelson Logic.