Theorem nfnth 1835

Description: No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) df-nf 1817 changed. (Revised by Wolf Lammen, 12-Sep-2021.)

Hypothesis

Ref	Expression
nfnth.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
nfnth	⊢ Ⅎ𝑥𝜑

Proof of Theorem nfnth

Step	Hyp	Ref	Expression
1		nfntht2 1827	. 2 ⊢ (∀𝑥 ¬ 𝜑 → Ⅎ𝑥𝜑)
2		nfnth.1	. 2 ⊢ ¬ 𝜑
3	1, 2	mpg 1830	1 ⊢ Ⅎ𝑥𝜑

This proof depends on syntax axioms:  ¬ wn 3  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by:  nffal  1838  nd1  10590  nd2  10591