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Metamath Proof Explorer

Theorem 19.3t

Description: Closed form of 19.3 and version of 19.9t with a universal quantifier. (Contributed by NM, 9-Nov-2020) (Proof shortened by BJ, 9-Oct-2022)

		Ref	Expression
	Assertion	19.3t	⊢ ( Ⅎ 𝑥 𝜑 → ( ∀ 𝑥 𝜑 ↔ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		sp	⊢ ( ∀ 𝑥 𝜑 → 𝜑 )
2		nf5r	⊢ ( Ⅎ 𝑥 𝜑 → ( 𝜑 → ∀ 𝑥 𝜑 ) )
3	1 2	impbid2	⊢ ( Ⅎ 𝑥 𝜑 → ( ∀ 𝑥 𝜑 ↔ 𝜑 ) )