Theorem 19.9h

Description: A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of Margaris p. 89. (Contributed by FL, 24-Mar-2007) (Proof shortened by Wolf Lammen, 5-Jan-2018) (Proof shortened by Wolf Lammen, 14-Jul-2020)

	Ref	Expression
Hypothesis	19.9h.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
Assertion	19.9h	⊢ ( ∃ 𝑥 𝜑 ↔ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		19.9h.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2	1	nf5i	⊢ Ⅎ 𝑥 𝜑
3	2	19.9	⊢ ( ∃ 𝑥 𝜑 ↔ 𝜑 )