The image you provided shows a symbolic expression and related premises for a logical argument. Based on the content, the task seems to involve proving the expression using rules of inference.

Here’s a breakdown of the symbols in the given expression and premises:

Expression (D): $[(p \rightarrow q) \land (q \rightarrow r) \land (p \lor s) \land (\neg t)] \rightarrow (\neg s \land \neg t)$

Premises:

• P1: $p \rightarrow q$
• P2: $q \rightarrow r$
• P3: $p \lor s$
• P4: $\neg t$
• P5: $\neg t$

Table of Inference: The table shows that different inference laws are applied to the premises to reach a conclusion. Here’s what each line suggests:

• Syllogism (P2, P3): Using a combination of $p \rightarrow q$ and $q \rightarrow r$ to infer a conclusion.
• Hypothetical (P6, P3): Based on specific inference laws, using premises $P6$ and $P3$.
• Modus Tollens (MTT) for negations.
• Simplification and Adjunction are other inference rules applied to reduce or combine premises.

Would you like help walking through the proof process, or do you have a specific step you need assistance with?