Truth table and logic question

Question

Let $p,q$ and $r$ be sentances

I know that $[(p\to q)\land (p\to r)]\iff p\to(q\land r)$. (1) is true.

Let $p,q,r,s,t$ be sentences

Is $[(p\to q)\land (p\to r)\land (p\to s)\land (p\to t)]\iff p\to(q\land r\land s\land t)$ be true by (1)

If not what would I do to make it a tautology?

Btw:What is the mathjax for union,intersection

Answer 1

It is valid.

Take the LHS as premise, then assuming $p$ enables the collective derivation of all of the conjuncts ($q,r,s,t$). Therefore the LHS entails the RHS.

Take the RHS as premise, then assuming $p$ multiple times enables the individual derivation of each of the conjuncts ($q,r,s,t$). Therefore the RHS entails the LHS.

Therefore the LSH entails and is entailed by the RHS, so it is a valid equivalence.

And you can prove this by a truth table.

$$\boxed{\begin{array}{|c|c|}p&q&r&s&t&(p\to q)\wedge(p\to r)\wedge (p\to s)\wedge p\to t)&p\to(q\wedge r\wedge s\wedge t)\\\hline\top&\top&\top&\top&\top&\top&\top\\\top&\top&\top&\top&\bot&\bot&\bot\\\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\end{array}}$$

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PS:

$\land, \lor, \lnot, \to, \implies, \iff$ gives: $\land, \lor, \lnot, \to, \implies, \iff$

also $\leftrightarrow$ gives $\leftrightarrow$