Theorem 19.9

Description: 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)

	Ref	Expression
Hypothesis	19.9.1	$$⊢Ⅎx\phi$$
Assertion	19.9	$$⊢\exists x\phi \leftrightarrow \phi$$

Proof

Step	Hyp	Ref	Expression
1		19.9.1	$$⊢Ⅎx\phi$$
2		19.9t	$$⊢Ⅎx\phi \to \left(\exists x\phi \leftrightarrow \phi \right)$$
3	1 2	ax-mp	$$⊢\exists x\phi \leftrightarrow \phi$$