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Theorem bj-nnf-alrim 34103
Description: Proof of the closed form of alrimi 2215 from modalK (compare alrimiv 1929). See also bj-alrim 34045. Actually, most proofs between 19.3t 2203 and 2sbbid 2249 could be proved without ax-12 2179. (Contributed by BJ, 20-Aug-2023.)
Assertion
Ref Expression
bj-nnf-alrim (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))

Proof of Theorem bj-nnf-alrim
StepHypRef Expression
1 bj-nnfa 34079 . 2 (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
2 alim 1812 . 2 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
31, 2syl9 77 1 (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  Ⅎ'wnnf 34074
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 34075
This theorem is referenced by:  bj-stdpc5t  34116  bj-19.21t  34117
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