Theorem nfri 1789

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

Hypothesis

Ref	Expression
nfri.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfri	⊢ (∃𝑥𝜑 → ∀𝑥𝜑)

Proof of Theorem nfri

Step	Hyp	Ref	Expression
1		nfri.1	. 2 ⊢ Ⅎ𝑥𝜑
2		df-nf 1784	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
3	1, 2	mpbi 230	1 ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1538  ∃wex 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-nf 1784

This theorem is referenced by:  nf5ri  2196