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The procedure for modifying column values in a pandas dataframe by matching them with the column values of another dataframe is as follows:

1. Import the pandas library and read both dataframes.

   import pandas as pd
   
   df1 = pd.read_csv('file1.csv')
   df2 = pd.read_csv('file2.csv')
   

2. Ensure that the columns to be matched are of the same data type in both dataframes.

   df1['column'] = df1['column'].astype(str)
   df2['column'] = df2['column'].astype(str)
   

3. Create a new column in the first dataframe to store the modified values.

   df1['new_column'] = df1['column']
   

4. Use a for loop to iterate through each row in the first dataframe.

   for index, row in df1.iterrows():
   

5. Within the for loop, use the .loc[] function to match the value in the current row of the first dataframe with the corresponding value in the second dataframe. Then modify the value in the new column of the first dataframe accordingly.

   corresponding_value = df2.loc[df2['column'] == row['column']]['new_column'].values[0]
   
   df1.loc[index, 'new_column'] = corresponding_value
   

6. Finally, save the modified dataframe to a new file.

   df1.to_csv('modified_file.csv', index=False)