A lion is a cat - Lew TO kot [lew (nominative case) = kot (nominative)]
The sentence "Lew TO kot" means: "A lion" = "a cat" (the same thing)
A lion IS a cat - Lew JEST kotem [lew (nom) JEST kotem (instrumental)]
The sentence "Lew JEST kotem" uses the verb JEST to define, to describe
the lion, and requires that descriptive noun be in the instrumental case.
Lew JEST psem (instrumental)
Lew nie JEST psem (instrumental)
A different angle: is TO more like the translation is, in the context of existence (sort of a passive understanding of is), whereas JEST is more along the lines of a state of being (in an active sense)? Therefore instrumental is needed after JEST because it has to describe what a person/animal/thing is being? As opposed to TO which is just a statement — “This is ____.”
If I would reword it, I might say Lew nie jest psem. A lion is not (in the state of being) a dog.
Lew to nie pies. A lion is not a dog. (Statement of fact. Or one could say, this means the definition of a lion is not compatible with that of a dog. Therefore A lion ≠ a dog.)
An alternative implication for JEST could be that a lion is not (continuing - in the sense of an ongoing state - to be) a dog.
If one could respond to this comment, so I know that I am considering the differentiations in these set phrases correctly, that would be immensely helpful. Dzięnkuję!!!
to be - być
it is - JEST
Instrumental - Narzędnik (kim? czym?) JEST psem/Nie JEST psem
This case is used to define, or to describe thing(s) in a full sentence.
The verb JEST puts the noun in direct object and needs the instrumental:
Lew (nominative) JEST psem (instrumental) - A lion IS a dog
Lew (nominative) nie JEST psem (instrumental) - A lion IS not a dog
In the phrase: "Lew TO (jest) pies" - "jest" goes to the back of your head.
The "TO" gives enough information: a lion (subject) = a dog (same thing)
Therefore, "TO" behaves like the English predicate nominative, which refers
back to the subject (nominative) after the verbs "to be" and "to become":
Lew TO pies (nominative) - A lion (nominative) is a dog (nominative)
Lew TO nie pies (nominative) - A lion (nominative) is not a dog (nominative)
The construction "Lew to pies" works as if there was some kind of an invisible 'jest' in there. In fact, while it's not as common, it's still okay to write "Lew to jest pies".
So when you negate "Lew jest psem", you negate the whole idea of 'being a dog' and end up at "Lew nie jest psem". You should do the same with "Lew to (jest) pies" - you negate the whole idea of 'being a dog', including that invisible 'jest', and end up at "Lew to nie (jest) pies".