Just in you were wondering:
In classical logic "true implies false" implies "false implies true", but not conversely.
(Artwork due to Mike Stay.)
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@johncarlosbaez The symbol => is used by different people to mean either
-> (material implication, a logical connective)
or
|= (semantic entailment, "true in every model")
I understand you intend it the first sense, like in the 1st line in attached image. However, I was taught to use => in the second sense so I can't help to try to interpreted that way. I think that sort of works too---the 2nd and 3rd line in the image, with the middle => interpreted as the mathematicians' informal "if ... then".
