Theorem bj-nnfv 37111

Description: A non-occurring variable is nonfree in a formula. (Contributed by BJ, 28-Jul-2023.)

Assertion

Ref	Expression
bj-nnfv	⊢ Ⅎ'𝑥𝜑

Distinct variable group:   𝜑,𝑥

Proof of Theorem bj-nnfv

Step	Hyp	Ref	Expression
1		ax5e 1919	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
2		ax-5 1917	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3		df-bj-nnf 37070	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
4	1, 2, 3	mpbir2an 717	1 ⊢ Ⅎ'𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1545  ∃wex 1786  Ⅎ'wnnf 37069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-5 1917

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-bj-nnf 37070

This theorem is referenced by:  bj-pm11.53a  37113