Help finishing a logic proof?

Question

I have been working on the following proof:

Every fetus has an immortal soul. A thing has an immortal soul only if it has a right to life. Hence, every fetus has a right to life. (Fx = x is a fetus, Sx = x has an immortal soul, Rx = x has a right to life).

(x)(Fx → Sx) (x)(Rx → Sx) ∴ (x)(Fx → Rx)

I believe I got the symbolization correct, but if I did not please someone let me know. The proof I have up to this point is as follows:

(x)(Fx → Sx)
(x)(Rx → Sx) ∴ (x)(Fx → Rx)
Fa Assume (for CP)
Fa → Sa 1, UI
Sa 4, 3, MP
Ra → Sa 2, UI

At this point I was working on a conditional proof but I need to get "Ra" out of this one and I can't figure out the next step. Any help would be appreciated.

Answer 1

Politics/religion/morality aside...$P$ only if $Q$ translates to $P\rightarrow Q$

Answer 2

Here is an example I often use:

'You can be a bachelor only if you are male'

Using $B$ for 'you are a bachelor', and $M$ for 'you are male', should we translate this as:

$$M \rightarrow B$$

or

$$B \rightarrow M$$

I hope it is clear that the second translation is what we want: the first one ends up saying that you are a bachelor as soon as we know you are male, but of course that is not the case: you also have to be unmarried. But what is the case is that when you tell me that you are a bachelor, I then also know you are a male, since only males can be bachelors.

And so yes, in general, '$P$ only if $Q$' translates as:

$$P \rightarrow Q$$