Theorem bj-nnf-alrim 37739

Description: Proof of the closed form of alrimi 2249 from modalK (compare alrimiv 1959). See also bj-alrim 37687. Actually, most proofs between 19.3t 2237 and 2sbbid 2282 could be proved without ax-12 2212. (Contributed by BJ, 20-Aug-2023.)

Assertion

Ref	Expression
bj-nnf-alrim	⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))

Proof of Theorem bj-nnf-alrim

Step	Hyp	Ref	Expression
1		bj-nnfa 37722	. 2 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
2		alim 1842	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
3	1, 2	syl9 78	1 ⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  Ⅎ'wnnf 37720

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-4 1841

This proof depends on definitions:  df-bi 210  df-an 402  df-bj-nnf 37721

This theorem is used by:  bj-stdpc5t  37740  bj-19.21t  37755