Theorem bj-nnf-alrim 34199

Description: Proof of the closed form of alrimi 2211 from modalK (compare alrimiv 1928). See also bj-alrim 34140. Actually, most proofs between 19.3t 2199 and 2sbbid 2245 could be proved without ax-12 2175. (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 34175	. 2 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
2		alim 1812	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
3	1, 2	syl9 77	1 ⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))

Syntax hints:   → wi 4  ∀wal 1536  Ⅎ'wnnf 34170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-4 1811

This theorem depends on definitions:  df-bi 210  df-an 400  df-bj-nnf 34171

This theorem is referenced by:  bj-stdpc5t  34212  bj-19.21t  34213