Theorem bj-nnfnth 37433

Description: A variable is nonfree in the negation of a theorem, inference form. (Contributed by BJ, 27-Aug-2023.)

Hypothesis

Ref	Expression
bj-nnfnth.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
bj-nnfnth	⊢ Ⅎ'𝑥𝜑

Proof of Theorem bj-nnfnth

Step	Hyp	Ref	Expression
1		bj-nnfnth.1	. . 3 ⊢ ¬ 𝜑
2	1	bj-nnfth 37426	. 2 ⊢ Ⅎ'𝑥 ¬ 𝜑
3		bj-nnfnt 37432	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑)
4	2, 3	mpbir 234	1 ⊢ Ⅎ'𝑥𝜑

This proof depends on syntax axioms:  ¬ wn 3  Ⅎ'wnnf 37408

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-bj-nnf 37409

This theorem is used by: (None)