In Mendelson's book, Introduction to Mathematical Logic I put Lemma $2.29$, part $d$) in question. It reads $$ \vdash\neg ( \mathscr{D} \Rightarrow (\exists x)\mathscr{C} (x)) \iff (\exists y)( \mathscr{D} \Rightarrow \mathscr{C} (y)) $$ I don't understand what "$\vdash$" is for, but let me rewrite it like this $$ \neg ( D \Rightarrow \exists x\ C (x)) \iff \exists y\ ( D \Rightarrow C (y)) \tag{*} $$ I have a big problem with the negation sign at the begining. I could "derive" other parts of the same Lemma $2.29$, e.g. $b$) here in simple way, call it intuitive. This part here is not all right. could check it please.
This is my attempt, assuming that $\exists y P(y)$ can be understood as $\bigvee_y P(y):= P(y_1) \vee P(y_2) \vee \dots$, but let us take two disjuncts. Starting with the LHS of (*)
$$ \exists y\ ( D \Rightarrow P (y)) \\ \bigvee_y ( D \Rightarrow P(y))\\ (D \Rightarrow P(y_1) \vee (D \Rightarrow P(y_2)) \\ (\neg D \vee P(y_1)) \vee (\neg D \vee P(y_2)) \\ \neg D \vee P(y_1) \vee \neg D \vee P(y_2) \\ \neg D \vee P(y_1) \vee P(y_2) \\ \neg D \vee (P(y_1) \vee P(y_2)) \\ D \Rightarrow(P(y_1) \vee P(y_2)) \\ D \Rightarrow (\bigvee_y P(y)) \\ D \Rightarrow (\exists y\ P(y)) \\ $$ as you can see there is no $\neg$ at the end!. What is wrong ? Even in the 6th edition, Lemma $2.29$ was not changed.
The following image is from the 5th edition
