Theorem bj-nnfa 37381

Description: Nonfreeness implies the equivalent of ax-5 1939. See nf5r 2229. (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 37380	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
2	1	simprbi 502	1 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1808  Ⅎ'wnnf 37379

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 401  df-bj-nnf 37380

This theorem is used by:  bj-nnfad  37382  bj-nnfai  37383  bj-nnfea  37387  bj-nnfim1  37394  bj-nnfim2  37395  bj-nnf-alrim  37398  bj-19.23t  37415  bj-19.37im  37417  bj-19.42t  37418  bj-sbft  37431