Theorem nfrd

Description: Consequence of the definition of not-free in a context. (Contributed by Wolf Lammen, 15-Oct-2021)

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

Proof

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