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--- sav08:substitution_theorems_for_propositional_logic [2008/03/06 13:26]
vkuncak
+++ sav08:substitution_theorems_for_propositional_logic [2008/03/11 14:57]
vkuncak
@@ Line 38: / Line 38: @@
 ++++
-Corollary: if $\models F$, then $\models (subst(\sigma)(F))$ for every substitution $\sigma$.
+Corrollary: if $I_1(p) = I_2(p)$ for every $p \in FV(F)$, then $e(F)(I_1)=e(F)(I_2)$.
-We say that two formulas are equivalent if $\models (F \leftrightarrow G)$.
+Corollary: if $\models F$, then $\models (subst(\sigma)(F))$ for every variable substitution $\sigma$.
-**Lemma**: If $\models (F \leftrightarrow G)$ then for every interpretation $I$ we have $e(F)(I) = e(G)(I)$.
+Corollary: if $\models (F \leftrightarrow G)$ and $\sigma$ is a variable substitution, then
-**Theorem on instantiating equivalences**: if $\models (F \leftrightarrow G)$ and $\sigma$ is a variable substitution, then
 \[
     \models (subst(\sigma)(F) \leftrightarrow subst(\sigma)(G)
 \]
-Does the theorem hold if $\sigma$ is not variable substitution?
+++Does the theorem hold if $\sigma$ is not variable substitution?|No.++
+We say that two formulas are equivalent if $\models (F \leftrightarrow G)$.
+**Lemma**: If $\models (F \leftrightarrow G)$ then for every interpretation $I$ we have $e(F)(I) = e(G)(I)$.
 This theorem was about transforming formulas from the outside, ignoring the structure of certain subformulas.
@@ Line 57: / Line 61: @@
 **Theorem on Substituting Equivalent Subformulas**: if $\sigma$ is equivalence-preserving, then for every formula $F$ we have $\models (F \leftrightarrow (subst(\sigma)(F))$.