My class discussed the following theorem for which I wasn't able to make it to class. Its proof is supposed to involve structural induction but I am stuck in the inductive step...
Let B |=| C. If A' results from replacing some (not necessarily all) occurrences of B, in A, by C, then A |=| A'.
So far I have:
We want to prove R (the property above) for all formulas X. i.e. R(X) for all formulas X.
Base Case: X is an atomic formula (one variable) If X is the atomic formula x, and x |=| x', then X' is the formula x'. We conclude X |=| X'
Inductive Step: Assume R(X), and R(Y) for formulas X and Y. ???
The way I learned it, the inductive step involves assuming that some formulas have the property. We use this assumption to show that this implies
(X ^ Y), (X V Y), (X -> Y), and (-X)
i.e. R(X) ^ R(Y) -> (X ^ Y) ^ (X V Y) ^ (X -> Y) ^ (-X)
(Not sure about the above i.e., but that's what I think).
I'm having trouble with this step. Any help would be appreciated.
Thanks
