Theorem bj-nnfa 37552

Description: Nonfreeness implies the equivalent of ax-5 1943. See nf5r 2230. (Contributed by BJ, 28-Jul-2023.)

Assertion

Ref	Expression
bj-nnfa	⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Proof of Theorem bj-nnfa

Step	Hyp	Ref	Expression
1		df-bj-nnf 37551	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
2	1	simprbi 503	1 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812  Ⅎ'wnnf 37550

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-an 402  df-bj-nnf 37551

This theorem is used by:  bj-nnfad  37553  bj-nnfai  37554  bj-nnfea  37558  bj-nnfim1  37565  bj-nnfim2  37566  bj-nnf-alrim  37569  bj-19.23t  37586  bj-19.37im  37588  bj-19.42t  37589  bj-sbft  37602