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Metamath Proof Explorer

Theorem nf5r

Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021) (Proof shortened by Wolf Lammen, 23-Nov-2023)

		Ref	Expression
	Assertion	nf5r	⊢ ( Ⅎ 𝑥 𝜑 → ( 𝜑 → ∀ 𝑥 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		19.8a	⊢ ( 𝜑 → ∃ 𝑥 𝜑 )
2		id	⊢ ( Ⅎ 𝑥 𝜑 → Ⅎ 𝑥 𝜑 )
3	2	nfrd	⊢ ( Ⅎ 𝑥 𝜑 → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )
4	1 3	syl5	⊢ ( Ⅎ 𝑥 𝜑 → ( 𝜑 → ∀ 𝑥 𝜑 ) )