python toolbox display only the fields that are in both layer

[OrrGvili]

i have python tool box  with definition query tool, the tool activate definition query on multiple layers 

In running into hard wall when trying to display only the fields that mutual to the different layers 

 When try to display the list of the fields  the dialog shows only few fields and not all 

I'm testing it on 2 similar layers with 90 fields. Turned off 3 in one layer and the dialog show only 4  instead of 87 

With no filter i get all the fields in the first layer 

how to let the user see only the mutual 

list of fileds? 

    def updateParameters(self, parameters):
        """Modify the values and properties of parameters before internal
        validation is performed.  This method is called whenever a parameter
        has been changed."""
        if parameters[0].values:
            all_layers_field_list = []
            for layer in parameters[0].values:
                field_names = [f.name for f in arcpy.da.Describe(layer.name)["fields"]]
                all_layers_field_list.append(field_names)

        list_fields = list(set(all_layers_field_list[0]).intersection(*all_layers_field_list[1:]))
        parameters[1].filter.list = list_fields

        if parameters[1].value:
            field_values =list(set([row[0] for row in arcpy.da.SearchCursor(parameters[0].values[0], [parameters[1].valueAsText],f"{parameters[1].valueAsText} IS NOT NULL")]))
            parameters[2].filter.list = field_values             
        return

[DanPatterson]

I don't know what your *all_layers.... is all about 

a = list("abcdefg")

b = list("bcdfghij")

a
['a', 'b', 'c', 'd', 'e', 'f', 'g']

b
['b', 'c', 'd', 'f', 'g', 'h', 'i', 'j']

set(a).intersection(b)
{'b', 'c', 'd', 'f', 'g'}

set(a).intersection(*b)
set()

[OrrGvili]

 *all_layer_field_list its list containg the list of fields of eace layer 
the intesection works the problem the list of fields in the drop down  dont contain the full list 

[DanPatterson]

I was pointing out that the starred expression (line 14) yielded erroneous results

[OrrGvili]

im working with list of list becuse i dont know how many layer the user works with 

this woks with the list of list 

a = list("abcdefg")

b = list("bcdfghij")

d = [a,b]

set(d[0]).intersection(*d)

{'b', 'c', 'd', 'f', 'g'}

or with list of set its that way:  

set.intersection(*d)