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Metamath Proof Explorer

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	⊢ Ⅎ 𝑥 𝜑
	Assertion	19.9	⊢ ( ∃ 𝑥 𝜑 ↔ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		19.9.1	⊢ Ⅎ 𝑥 𝜑
2		19.9t	⊢ ( Ⅎ 𝑥 𝜑 → ( ∃ 𝑥 𝜑 ↔ 𝜑 ) )
3	1 2	ax-mp	⊢ ( ∃ 𝑥 𝜑 ↔ 𝜑 )