How to prove that the following conclusion is valid?

Question

With the help of premises: $(x)(P(x) \to Q(x)) , (x)(Q(x) \to R(x))$

Prove that $(x)(P(x) \to R(x))$ is a valid conclusion.

Question: (https://i.stack.imgur.com/soM22.jpg)

Edit: This is what I did (https://i.stack.imgur.com/glW9B.jpg)

Is this correct? Is it the correct way to derive the 3rd and 4th steps? Thank you

Answer 1

(Presuming your notation is a universal quantifier.)

Lines 3 and 4 are okay.   You are instantiating to the same arbitrary value, which is allowed.   (Rather than reuse $x$, I'd introduce a new token, but that's a style choice.)

Universal generalisation will take line 5 straight to line 7.

$\begin{array}{r|l:l} 1 & \langle x\rangle~(P(x)\to Q(x)) &\text{Premise}\\ 2 & \langle x\rangle~(Q(x)\to R(x)) & \text{Premise}\\ 3 & \quad P(a)\to Q(a) & 1, \text{Universal Instantiation }[x\backslash a] \\ 4 & \quad Q(a)\to R(a) & 2, \text{Universal Instantiation }[x\backslash a] \\ 5 & \quad P(a)\to R(a) & 3,4, \text{Hypothetical Syllogism}\\ \not 6~7 & \langle x\rangle~(P(x)\to R(x)) & 5,\text{Universal Generalisation } \end{array}$

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The hypothetical syllogism can be subproven, if needed.   Just assume $P(a)$, use conditional elimination (aka modus ponens) twice from lines 3,4, and then conditional introduction (aka deduction theorem) gives line 5.

Answer 2

Different formal proof systems can differ in how exactly they define specific inference rules, and so without you telling us what exactly the rules are of the system that you have to work with, we cannot tell whether you did this correctly ... but probably your steps 3 and 4 are correct.

However, I highly doubt your step 6 (though valid) is in line with however your UG rule is defined: typically, a universal generalization rule like that places a universal quantifier in front of the whole statement, not just par of it. However, as such UG lets you derive step 7 directly from step 5 though, so you can just delete line 6, and go from 5 to 7 directly by UG.