section{*Examples of Classical Reasoning*}
theory FOL_examples imports FOL begin
lemma "EX y. ALL x. P(y)-->P(x)"
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (rule exCI)
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (rule allI)
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (rule impI)
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (erule allE)
--{* @{subgoals[display,indent=0,margin=65]} *}
txt{*see below for @{text allI} combined with @{text swap}*}
apply (erule allI [THEN [2] swap])
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (rule impI)
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (erule notE)
--{* @{subgoals[display,indent=0,margin=65]} *}
apply assumption
done
text {*
@{thm[display] allI [THEN [2] swap]}
*}
lemma "EX y. ALL x. P(y)-->P(x)"
by blast
end