src/ZF/Constructible/MetaExists.thy

+header{*The meta-existential quantifier*}
+
+theory MetaExists = Main:
+
+text{*Allows quantification over any term having sort @{text logic}.  Used to
+quantify over classes.  Yields a proposition rather than a FOL formula.*}
+
+constdefs
+  ex :: "(('a::logic) => prop) => prop"            (binder "?? " 0)
+  "ex(P) == (!!Q. (!!x. PROP P(x) ==> PROP Q) ==> PROP Q)"
+
+syntax (xsymbols)
+  "?? "        :: "[idts, o] => o"             ("(3\<Or>_./ _)" [0, 0] 0)
+
+lemma meta_exI: "PROP P(x) ==> (?? x. PROP P(x))"
+proof -
+  assume P: "PROP P(x)"
+  show "?? x. PROP P(x)"
+  apply (unfold ex_def)
+  proof -
+    fix Q
+    assume PQ: "\<And>x. PROP P(x) \<Longrightarrow> PROP Q"
+    from P show "PROP Q" by (rule PQ)
+  qed
+qed 
+
+lemma meta_exE: "[| ?? x. PROP P(x);  !!x. PROP P(x) ==> PROP R |] ==> PROP R"
+apply (unfold ex_def)
+proof -
+  assume QPQ: "\<And>Q. (\<And>x. PROP P(x) \<Longrightarrow> PROP Q) \<Longrightarrow> PROP Q"
+  assume PR: "\<And>x. PROP P(x) \<Longrightarrow> PROP R"
+  from PR show "PROP R" by (rule QPQ)
+qed
+
+end