Rigorous Statements: "It suffices to show that [...]" and Variations Mathematical proofs display a variety of proof styles and language used. One of the common statements I have seen are "It suffices to show that [...]" and "We want to show that [...]". Of course, there are a few ways to say this: 


*

*We want to show that $x=y$. 

*It is desired to prove that $x=y$. 

*It suffices to show that $x=y$. 

*Our goal is to prove that $x=y$. 

*etc. 


I have heard many opinions and a lot of advice on this mentioning that one form of this statement is more rigorous than the other. General consensus tells me that graders and solvers find "it suffices to show that $x=y$" much more rigorous and somewhat more accurate than other forms of the same statement. Now, my question is, primarily, why? My other argument is the following. 
Say the problem states that we want to prove that $x=y$. We introduce a lemma in a proof. And, after that, we want to say "But we want to show that $x=y$". In such a case, "it suffices to show that $x=y$" is not appropriate, since that was the original statement of the problem itself. Your views? 
Note: This is not a soft question, because I am asking specifically about mathematical statements in a proof, which is a huge part of rigor and often writing skills score points on the olympiad. So, please don't misinterpret this as a soft question.
 A: The phrases you use do NOT mean the same thing, and they do NOT have different degrees of RIGOR.
I strongly suggest that you do not use words and phrases at all if you are not familiar with their meaning. "It suffices" means "It is enough" or "If we prove ... we are done (because of ...)". No reason to use "suffice" if you are not comfortable with it. You should be concerned with meaning not with "what graders and solvers find more rigorous". 
"It suffices to show $x=y$." means "It is ENOUGH to show $x=y$.", that is, you are saying that the proof of the original question (or the lemma whose proof you are writing) will be done once you have proved $x=y$.
Since this is a mathematical claim, this should be either reasonably obvious or follow from arguments immediately before the statement. Just using the word to be rigorous but not giving an argument gives a very bad impression of cargo cult proof writing. Many graders would severely punish this.
"We want to show that $x=y$./Our goal is to prove $x=y$."
Here, you are certainly not claiming that the proof of this identity will finish the proof. You are saying "I am going to proof $x=y$. Bear with me, I will explain later why this is useful.". Certainly, it is "less rigorous" than the "suffice" sentence if "suffice" is what you actually meant. But this is not the fault of the phrase. But see my last paragraph for better options.
"It is desired to prove that $x=y$."
This is not so good, because the passive construction makes it unclear if you say that YOU decided to prove $x=y$ because it fits into your brilliant proof strategy or if you think that the people who posed the problem wanted you to prove this. Often, this distinction is not so important, but it obfuscates your proof structure.
Finally: For a rigorous proof, you need to indicate the STRUCTURE of your proof. For a readable proof, you need to indicate the structure of your proof at the beginning and not at the end. The phrases you called "less rigorous" do not indicate proof structure. This means that you have to add information, it does not mean that the phrases themselves are not rigorous or that the other phrase would be better (indeed, the other phrase could be simply wrong).
So, for example, instead of just "We want to show that $x=y$.", it would be better to say "As a first step, we prove $x=y$." or "Next, we prove that $x=y$ because this will help us to prove that $z$ is even." or "Lemma 3: $x=y$". I personally favor the "Lemma/Fact/Step" approach because it automatically highlights your proof structure, but again, it is even better if you explain it.
