I am perplexed. Your first line alone is sufficient to deduce your third line.
[MATH]P \implies (\neg Q) \land R) \\
P \implies R.[/MATH]
If X is a dog, then X is not a snake and X has legs means that if X is a dog, it has legs.
And what does any of this have to do with modus ponens, which goes
[MATH]J \implies K \\
J \\
\therefore K.[/MATH]
Perhaps you are trying to go
[MATH]P \implies \neg Q \land R \\
and \ Q \implies P \lor R \\
\therefore Q \implies R.[/MATH]
I do not have a name for that argument, but it looks like it uses modus tollens
[MATH]\{P \implies \neg Q \land R\} \land \{Q \implies P \lor R\}.\\
\\
\{P \implies \neg Q \land R\} \implies \\
\{\neg \{\neg Q \land R\} \implies \neg P\} \implies \\
\{Q \lor \neg R \implies \neg P\} \implies \\
\{Q \implies \neg P\}
\\
But \ \{Q \implies P \lor R\} \\
\therefore Q \implies R.[/MATH]
If X is a dog, then X is not a snake and X has legs
If X is a snake or X has no legs (whether because X is a snake or because X is an oyster), then X is not a dog.
If X is a snake, then X is a dog or X has no legs. True, but we can simplify to if X is a snake, then X has no legs.
That argument seems to be a very complicated way of saying
If A entails B or C
and
If A entails not B,
then A entails C.
