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Quite the contrary - fascism arose in opposition to socialism.
Nope!
Mussolini was the publisher of Avanti! (Italy's leading socialist magazine). As Italy was being caught up in a wave of nationalism, a split emerged in the socialist party. They threw him out of it for supporting Italy's involvement in WWI - they wanted to stay out of it, he viewed it as a way to strike a blow against the anti-socialist monarchies of the Habsburg in Austria-Hungary and the Hohenzollerns in Germany. He also believed that the inevitable mobilization of Russian conscripts would destabilize the cazrist regime in Russia and hasten its move towards socialism (and he was right).
Mussolini felt that Italian socialists had therefore failed the original Marxian notion of a worldwide Revolution of the Proletariat and that WWI offered the opportunity for radical change around the globe.
After they threw him out he decided that he needed to go another way. In 1914 he formed Il Popolo d'Italia an interventionalist newspaper and started taking money from corporate clients - armament makers - and foreign intervention-minded socialists alike (who wanted Italy in the war on the allies' side).
Considering the nationalist fervor, he then decided that the nation-state wasn't something that class couldn't transcend, writing
The nation has not disappeared. We used to believe that the concept was totally without substance. Instead we see the nation arise as a palpitating reality before us! ... Class cannot destroy the nation. Class reveals itself as a collection of interests—but the nation is a history of sentiments, traditions, language, culture, and race. Class can become an integral part of the nation, but the one cannot eclipse the other.
and
The class struggle is a vain formula, without effect and consequence wherever one finds a people that has not integrated itself into its proper linguistic and racial confines—where the national problem has not been definitely resolved. In such circumstances the class movement finds itself impaired by an inauspicious historic climate.
At this point he decided to embark on a kind of a national socialist journey where anyone...regardless of class...could join in an advance the collective goals of the nation-state. With himself as the head of it, of course.
This where your statement of fascism rose to prominence in reaction to and in opposition to the Bolshevik Revolution of 1917 is both correct and incorrect. It's correct in that after the war he looked at Bolshevism's rise in the new Soviet Union as a threat to his idea of the nation-state's preeminence because Lenin was a proponent of the classical Marxian idea of a worldwide socialist revolution. It's incorrect to say that fascism's only rise was due to that because as we've shown, he'd begun moving in this direction as early as 1914 when the war started. In fact, he formed the Fasci d'Azione Rivoluzionaria in 1914 and started calling them Fascisti that same year.
After the war Mussolini determined that socialism had failed and what was needed instead was a strongman to run Italy. At this point the Mare Nostrum and spazio vitale stuff started.