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Metamath Proof Explorer

Theorem nfri

Description: Consequence of the definition of not-free. (Contributed by Wolf Lammen, 16-Sep-2021)

Ref Expression
Hypothesis nfri.1 ⊢ Ⅎ 𝑥 𝜑
Assertion nfri ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 )

Proof

Step Hyp Ref Expression
1 nfri.1 ⊢ Ⅎ 𝑥 𝜑
2 df-nf ⊢ ( Ⅎ 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )
3 1 2 mpbi ⊢ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 )