Theorem bj-nnfnth 34072

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 34071	. 2 ⊢ Ⅎ'𝑥 ¬ 𝜑
3		bj-nnfnt 34069	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑)
4	2, 3	mpbir 233	1 ⊢ Ⅎ'𝑥𝜑

Syntax hints:  ¬ wn 3  Ⅎ'wnnf 34055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-bj-nnf 34056

This theorem is referenced by: (None)