Theorem nfi 1821

Description: Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Wolf Lammen, 15-Sep-2021.)

Hypothesis

Ref	Expression
nfi.1	⊢ (∃𝑥𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
nfi	⊢ Ⅎ𝑥𝜑

Proof of Theorem nfi

Step	Hyp	Ref	Expression
1		nfi.1	. 2 ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)
2		df-nf 1817	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
3	1, 2	mpbir 234	1 ⊢ Ⅎ𝑥𝜑

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-nf 1817

This theorem is used by:  nfv  1947  nfexhe  2214