Theorem 19.9

A wff may be existentially quantified with a variable not free in it. Version of 19.3 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v for a version requiring fewer axioms. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.)

Hypothesis
19.9.1
\( \vdash \Finv x \, \varphi \)
Conclusion
\( \vdash \exists x \, \varphi \leftrightarrow \varphi \)
Proof
l0
19.9.1
\( \vdash \Finv x \, \varphi \)
19.9t
\( \vdash \Finv x \, \varphi \rightarrow (\exists x \, \varphi \leftrightarrow \varphi) \)
ax-mp l0 _