NOPE!
the House acts as prosecutors and grand juries, they investigate and decide if high crimes and misdemeanors have met a ''probable cause'' burden needed, in order to place charges.
the Senate tries the defendant, acting as jurors and judges.
the Senate is where all evidence supporting the charges is presented and given to them, and all exculpatory evidence the defense team may have, gets presented and turned over as well.There is no rule that it has to only be evidence the House collected before the impeachment vote.
(no prosecutor stops collecting evidence after he has charged a person, he continues to collect new evidence and submits it thru the trial)
the Senate has a much greater burden than the House with their votes, because the senators can only convict with the standard of ''BEYOND a reasonable doubt'' not simply a ''probable cause'' like the House.
the trial stage is where even more evidence needs presented in order to reach that much higher burden of proof, to convict.
Holding material evidence back is not what happens in trials, presenting all the evidence you got and some, is needed by both sides, in order for Senators to get to the truth, and make their impartial, fair, just, and sound verdict.
That's all very nice, but is any of it true?
The only provisions found are, the House has the sole power to impeach, and the Senate has the sole power to try all impeachments. Just about the only rule is, there's a super-majority required to convict. That is it. Just as there is no rule there be only House-collected evidence in the trial, there is no rule as to the applicable evidentiary standards in either the House or the Senate. They can follow whatever standards they wish to apply, or no standards at all, provided they have the votes.
That is to say, the Senate is well within Constitutional language to acquit despite overwhelming evidence of guilt, despite the most egregious abuses of power, and also to convict based on the flimsiest of evidence for the most inconsequential personal foibles. That's just how it is. Just about the only remedy for either is an electorate sufficiently dissatisfied with either course to throw the bums out the next opportunity they get. That's the meaning of the phrase, "The Senate is also on trial."
So, what we are seeing in the Senate is not a trial analogous to the proceedings of a court of law, but a fight before the court of public opinion for the public support for the respective sides' judgment in light of "evidence" provided, or narratives promulgated, as the case may be. The claim to be in search of "evidence" - in the form of documents or witness testimonies - is one way to go about it, hinting at a trial before a court of law analogy, just as is the denigration of, and the lying about, the House procedures. In the end, what Schiff & Co. are doing is not pursuing a "beyond a reasonable doubt" standard, but to make the clown show on behalf of the president as unreasonable and unsupportable as it actually is, so as to convince as many as possible out of the Goobers' electorate that the Senate majority will vote to acquit because they are as corrupt as is Trump. Because, according to every reasonable standard, and by their overwhelming majority, they are.